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massive update for Mogan Research 1.2.5
2024-02-24 11:38:26 +08:00
<TeXmacs|2.1.2>
加入新课件：微积分第二章
2022-10-03 00:40:17 +08:00
<style|<tuple|projector|reddish|framed-theorems|number-long-article|invisible-multiply>>
<\body>
<\hide-preamble>
<assign|math|<macro|x|<math-colored|<with|math-display|true|<uncolored-math|<arg|x>>>>>>
</hide-preamble>
<screens|<\shown>
<chapter*|Chapter 2: Limits and Derivatives>
<section|The Tangent and Velocity Problems>
<subsection|The tangent problem (how to find the tangent line)>
\;
UIC: 移除微积分第二章中的color pattern
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<with|gr-mode|<tuple|group-edit|edit-props>|gr-frame|<tuple|scale|1cm|<tuple|0.28001gw|0.499999gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-fill-color|red|gr-color|blue|<graphics||<spline|<point|-2.95384|0.5>|<point|-0.7|-0.7>|<point|2.0|1.7>>|<math-at|y=f<around*|(|x|)>|<point|2.2|1.7>>|<with|color|red|fill-color|red|<point|-0.4|-0.602261>>|<with|color|red|<text-at|secant
加入新课件：微积分第二章
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line|<point|1.9|0.4>>>|<with|color|red|<math-at|<around*|(|a,f<around*|(|a|)>|)>|<point|-1.53969986357435|-0.260668072264252>>>|<with|color|red|<math-at|<around*|(|b,f<around*|(|b|)>|)>|<point|0.23470999214519|1.11374178345529>>>|<with|color|blue|fill-color|red|<line|<point|-0.4|-0.602261>|<point|1.6|0.3>>>|<with|color|red|fill-color|red|<line|<point|-0.4|-0.602261>|<point|1.47921610728005|1.0>>>|<with|color|red|<math-at|y=<frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>*<around*|(|x-a|)>+f<around*|(|a|)>|<point|2.0|-0.3>>>|<with|color|blue|fill-color|red|<\document-at>
\;
</document-at|<point|3.2|-2.5>>>|<with|color|blue|<\document-at>
UIC: 移除微积分第二章中的color pattern
2022-10-06 01:02:09 +08:00
In fact,
加入新课件：微积分第二章
2022-10-03 00:40:17 +08:00
UIC: 移除微积分第二章中的color pattern
2022-10-06 01:02:09 +08:00
<\equation*>
m=lim<rsub|b\<rightarrow\>a> <frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>
</equation*>
加入新课件：微积分第二章
2022-10-03 00:40:17 +08:00
</document-at|<point|3.7|-1.3>>>|<with|color|blue|<\document-at>
tangent line:
<\equation*>
y=m*x+f<around*|(|a|)>
</equation*>
\ for some slope m
UIC: 移除微积分第二章中的color pattern
2022-10-06 01:02:09 +08:00
</document-at|<point|0.3|-1.0>>>|<\with|fill-color|red|color|blue>
\;
</with>>>
加入新课件：微积分第二章
2022-10-03 00:40:17 +08:00
<subsection|The velocity problem (how to find instantaneous velocity)>
<\unfolded-plain>
Let <math|S<around*|(|t|)>> be the position of an object at time
<math|t>. The average velocity from <math|t=a> to <math|t=b> is
<\equation*>
<frac|S<around*|(|b|)>-S<around*|(|a|)>|b-a>
</equation*>
The instantaneous velocity is the limit of the average velocity in
shorter and shorter time intervals, i.e.
<\equation*>
lim<rsub|b\<rightarrow\>a> <frac|S<around*|(|b|)>-S<around*|(|a|)>|b-a>
</equation*>
<\remark>
Limit is always a dynamic process!
</remark>
<|unfolded-plain>
\;
</unfolded-plain>
<section|The Limit of a Function>
<subsection|Intuitive definition of a limit>
<\definition>
For a given function <math|f<around*|(|x|)>>, and a point <math|x=a>
<deleted|in the domain of <math|f>>, we say
<\equation*>
<with|color|blue|<with|color|red|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>=L>>,
</equation*>
read as: the limit of <math|f<around*|(|x|)>>, as <math|x> approaches
<math|a>, equals <math|L>, if <math|f<around*|(|x|)>> gets arbitrarily
close to <math|L> as long as <math|x> is sufficiently close to
<math|a>.
</definition>
<\example>
Let <math|f<around*|(|x|)>=x<rsup|2>>, then\
<\equation*>
lim<rsub|x\<rightarrow\>3> f<around*|(|x|)>=9
</equation*>
</example>
<\wide-block>
<tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
<\example>
Find the equation of the tangent line to the graph of
<math|f<around*|(|x|)>=x<rsup|2>> at <math|x=1>.
<\answer*>
Draw a secant line from <math|<around*|(|1,1|)>> to
<math|<around*|(|x,f<around*|(|x|)>|)>>. Its slope is
<\equation*>
m<around*|(|x|)>=<frac|f<around*|(|x|)>-1|x-1>=<frac|x<rsup|2>-1|x-1>.
</equation*>
Then the slope of the tangent line is
<\equation*>
lim<rsub|x\<rightarrow\>1> m<around*|(|x|)>=lim<rsub|x\<rightarrow\>1>
<frac|x<rsup|2>-1|x-1>=lim<rsub|x\<rightarrow\>1> x+1=2.
</equation*>
So the equation of the tangent line is
<\equation*>
y=2*<around*|(|x-1|)>+1=2*x-1.
</equation*>
</answer*>
</example>
</cell>>>>
</wide-block>
<\question>
\;
<\equation*>
lim<rsub|x\<rightarrow\>0> <frac|sin x|x>=?
</equation*>
Hint: draw a circle.
<with|gr-mode|<tuple|edit|math-at>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|2>|gr-grid-old|<tuple|cartesian|<point|0|0>|2>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|2>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|2>|gr-snap|<tuple>|gr-arrow-end|\<gtr\>|gr-color|blue|<graphics||<carc|<point|-2|0>|<point|2.0|0.0>|<point|0.0|-2.0>>|<with|color|blue|<line|<point|0|0>|<point|2.0|0.0>>>|<with|color|blue|<line|<point|0|0>|<point|1.73205080756888|1.0>>>|<with|color|blue|<math-at|x|<point|0.6|0.114858>>>|<with|color|blue|<math-at|r=1|<point|0.8|-0.300381>>>|<with|color|blue|<line|<point|1.73428|0.999297>|<point|1.7438009012361|0.00869816858902807>>>|<with|color|blue|arrow-end|\<gtr\>|<line|<point|2.34388|1.1517>|<point|1.7819008640291|0.427797759312084>>>|<with|color|blue|<math-at|sin
x|<point|2.55343|1.24695>>>|<with|arrow-end|\<gtr\>|<line|<point|2.91538|0.627823>|<point|2.02955062218364|0.294447889536566>>>|<math-at|x|<point|3.09636|0.637348>>|<with|color|blue|<math-at|lim<rsub|x\<rightarrow\>0>
<frac|sin x|x>=1|<point|2.34388|-0.705676>>>>>
</question>
<\example>
\;
<\equation*>
lim<rsub|x\<rightarrow\>0> sin <frac|1|x>=?
</equation*>
The limit does not exist!\
</example>
<\example>
\;
<\equation*>
lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=0
</equation*>
</example>
<with|ornament-border|1ln|ornament-color|red|ornament-shape|rounded|<\remark>
In the definition of <math|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>=L>, the function value <math|f<around*|(|a|)>> is
irrelevant. In fact, <math|f> can even be undefined on <math|x=a>.
</remark>>
<\example>
Let
<\equation*>
f<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1-x,>|<cell|x\<less\>0,>>|<row|<cell|e<rsup|x>,>|<cell|x\<gtr\>0,>>|<row|<cell|5,>|<cell|x=0,>>>>><space|1em>g<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1-x,>|<cell|x\<less\>0,>>|<row|<cell|e<rsup|x>,>|<cell|x\<gtr\>0.>>>>>
</equation*>
Find <math|lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>> and
<math|lim<rsub|x\<rightarrow\>0> g<around*|(|x|)>>.
<\answer*>
<\equation*>
lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>=1\<neq\>f<around*|(|0|)>
</equation*>
<\equation*>
lim<rsub|x\<rightarrow\>0> g<around*|(|x|)>=1,<space|1em>g<around*|(|0|)>
<text|is undefined>
</equation*>
</answer*>
</example>
<subsection|One-sided limits>
<\definition>
For a given function <math|f<around*|(|x|)>>, and a point <math|x=a>,
\ we say
<\equation*>
<with|color|blue|<with|color|red|lim<rsub|x\<rightarrow\><with|color|dark
green|a<rsup|->>> f<around*|(|x|)>=L>>,
</equation*>
read as: the <strong|left-hand limit> of <math|f<around*|(|x|)>>, as
<math|x> approaches <math|a>, equals <math|L>, if
<math|f<around*|(|x|)>> gets arbitrarily close to <math|L> as long as
<math|x\<less\>a> is sufficiently close to <math|a>.
Definition of the <strong|right-hand limit> is similar.\
</definition>
<\example>
Find the left and right hand limits of
<\equation*>
f<around*|(|x|)>=<choice|<tformat|<table|<row|<cell|x-1,>|<cell|x\<less\>0,>>|<row|<cell|sin
x+3,>|<cell|x\<gtr\>0.>>>>>
</equation*>
<\answer*>
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|->>
f<around*|(|x|)>=-1,<space|1em>lim<rsub|x\<rightarrow\>0<rsup|+>>
f<around*|(|x|)>=3.
</equation*>
</answer*>
</example>
\;
Relation between limit and one-sided limits:
<\itemize>
<item>If the limit exists, the both left and right hand limit exsits
and equal to the limit value itself
<item>If both left and right hand limit exists and equal to the same
value, then the limit exists and equal to the same value.
<item>If both left and right hand limit exists but they are not equal,
then the limit does not exist.\
</itemize>
\;
<\example>
\
<\wide-tabular>
<tformat|<cwith|1|1|2|2|cell-valign|c>|<cwith|1|1|1|1|cell-valign|c>|<table|<row|<\cell>
<image|<tuple|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
</cell>|<\cell>
<\itemize>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2<rsup|->>
g<around*|(|x|)>=3>>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2<rsup|+>>
g<around*|(|x|)>=1>>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>2>
g<around*|(|x|)>>> not exist!
<item><math|g<around*|(|2|)>> is undefined!
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|->>
g<around*|(|x|)>=2>>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|+>>
g<around*|(|x|)>=2>>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>5<rsup|>>
g<around*|(|x|)>=2>>
<item><math|g<around*|(|5|)>\<approx\>1.2>
<item><math|<with|math-display|true|lim<rsub|x\<rightarrow\>3>
g<around*|(|x|)>=1.5=g<around*|(|3|)>>>
</itemize>
</cell>>>>
</wide-tabular>
</example>
<subsection|Infinite limits>
<\definition>
We write
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=\<infty\>
</equation*>
if <math|f<around*|(|x|)>> gets arbitrarily large when <math|x> is
sufficiently close to <math|a>.
Similarly, we can define
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=-\<infty\>
</equation*>
</definition>
<\note*>
In these cases we will not say the limit exists.\
</note*>
<\example*>
<\equation*>
lim<rsub|x\<rightarrow\>0> <frac|1|x<rsup|2>>=\<infty\>
</equation*>
</example*>
<strong|One-sided infinite limits> can be defined similarly
<\equation*>
lim<rsub|x\<rightarrow\>a<rsup|->> f<around*|(|x|)>=\<infty\><infix-or>-\<infty\>
</equation*>
<\equation*>
lim<rsub|x\<rightarrow\>a<rsup|+>> f<around*|(|x|)>=\<infty\><infix-or>-\<infty\>
</equation*>
<\example*>
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|+>>
<frac|1|x>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>0<rsup|->>
<frac|1|x>=-\<infty\>.
</equation*>
</example*>
<\example*>
\;
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|+>> ln
x=-\<infty\>,<space|1em>lim<rsub|x\<rightarrow\><around*|(|<frac|\<pi\>|2>|)><rsup|->>
tan x=\<infty\>,<space|1em><space|1em>lim<rsub|x\<rightarrow\><around*|(|<frac|\<pi\>|2>|)><rsup|+>>
tan x=-\<infty\>
</equation*>
</example*>
<\definition>
If the left/right hand limit of <math|f<around*|(|x|)>> is
<math|\<infty\>> or <math|-\<infty\>> as <math|x> approaches <math|a>,
then the line <math|x=a> is called a vertical asymptote of <math|f>.\
</definition>
\;
<section|Calculating Limits Using the Limit Laws>
<\wide-block>
<tformat|<cwith|1|-1|1|-1|cell-lsep|2spc>|<cwith|1|-1|1|-1|cell-rsep|2spc>|<cwith|1|-1|1|-1|cell-bsep|2spc>|<cwith|1|-1|1|-1|cell-tsep|2spc>|<table|<row|<\cell>
<strong|Limit Laws >(for limit or one-sided limits)
<\enumerate>
<item>Sum Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
<around*|[|f<around*|(|x|)>+g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>+lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
provided the limits on the right-hand-side exist.\
<item>Difference Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
<around*|[|f<around*|(|x|)>-g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>-lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
provided the limits on the right-hand-side exist.\
<item>Constant Multiple Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
c*f<around*|(|x|)>=c*lim<rsub|x\<rightarrow\>a>> f<around*|(|x|)>>
provided the limit on the right-hand-side exists, where <math|c> is
a constant. \
<item>Product Law: <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
<around*|[|f<around*|(|x|)>*g<around*|(|x|)>|]>=lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>*lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>>>
provided the limits on the right-hand-side exist.\
<item>Quotient Law: \ <math|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
<frac|f<around*|(|x|)>|g<around*|(|x|)>>=<frac|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>>|<with|math-display|true|lim<rsub|x\<rightarrow\>a>
g<around*|(|x|)>>>>> provided the limits on the right-hand-side
exist and <math|lim<rsub|x\<rightarrow\>a>
g<around*|(|x|)>\<neq\>0>.\
</enumerate>
</cell>>>>
</wide-block>
\;
Generalize to any <strong|algebraic combination> of functions: e.g.\
<\equation*>
lim<rsub|x\<rightarrow\>a> 2*f<around*|(|x|)>+5*g<around*|(|x|)>=2*lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>+5*lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>.
</equation*>
Power Law: <math|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)><rsup|n>=<around*|[|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>|]><rsup|n>>.\
Root Law: <math|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)><rsup|1/n>=<around*|[|lim<rsub|x\<rightarrow\>a>
f<around*|(|x|)>|]><rsup|1/n>> (<math|n> must be odd if
<math|lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>\<less\>0>).\
Two simple limits:
<\equation*>
lim<rsub|x\<rightarrow\>a> c=c,<space|1em>lim<rsub|x\<rightarrow\>a>
x=a.
</equation*>
\;
<\wide-block>
<tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
<strong|Direct substitution property> for polynomials and rational
functions:
<\equation*>
lim<rsub|x\<rightarrow\>a> P<around*|(|x|)>=P<around*|(|a|)>,<space|1em>lim<rsub|x\<rightarrow\>a>
<frac|P<around*|(|x|)>|Q<around*|(|x|)>>=<frac|P<around*|(|a|)>|Q<around*|(|a|)>>,<text|if
>Q<around*|(|a|)>\<neq\>0.
</equation*>
where <math|P,Q> are polynomials.\
</cell>>>>
</wide-block>
\;
<\wide-block>
<tformat|<cwith|1|-1|1|-1|cell-lsep|1spc>|<cwith|1|-1|1|-1|cell-rsep|1spc>|<cwith|1|-1|1|-1|cell-bsep|1spc>|<cwith|1|-1|1|-1|cell-tsep|1spc>|<table|<row|<\cell>
<\example>
\;
<\equation*>
lim<rsub|x\<rightarrow\>2> <around*|(|x<rsup|3>-4x+5|)>=2<rsup|3>-4\<cdot\>2+5=5.
</equation*>
<\equation*>
lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-2x|x<rsup|5>+1>=<frac|1-2|2>=-<frac|1|2>.
</equation*>
<\equation*>
lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-2x|x<rsup|3>-1>=lim<rsub|x\<rightarrow\>1>
<frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>
<text|<space|1em>does not exist>
</equation*>
In fact
<\equation*>
lim<rsub|x\<rightarrow\>1<rsup|->>
<frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>1<rsup|+>>
<frac|x<around*|(|x-2|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=-\<infty\>
</equation*>
Another example:
<\equation*>
lim<rsub|x\<rightarrow\>1> <frac|x<rsup|2>-x|x<rsup|3>-1>=lim<rsub|x\<rightarrow\>1><frac|x<around*|(|x-1|)>|<around*|(|x-1|)><around*|(|x<rsup|2>+x+1|)>>=lim<rsub|x\<rightarrow\>1><frac|x|<around*|(|x<rsup|2>+x+1|)>>=<frac|1|3>.
</equation*>
</example>
</cell>>>>
</wide-block>
\;
<\wide-block>
<tformat|<cwith|1|1|1|1|cell-lsep|1spc>|<cwith|1|1|1|1|cell-rsep|1spc>|<cwith|1|1|1|1|cell-bsep|1spc>|<cwith|1|1|1|1|cell-tsep|1spc>|<table|<row|<\cell>
If <math|f<around|(|x|)>=g<around|(|x|)>> when <math|x\<neq\>a>, then
<math|lim<rsub|x\<rightarrow\>a> f<around|(|x|)>=lim<rsub|x\<rightarrow\>a>
g<around|(|x|)>>, provided the limits exist.
</cell>>>>
</wide-block>
\;
<\theorem>
If <math|f<around*|(|x|)>\<leqslant\>g<around*|(|x|)>> when <math|x> is
near <math|a> (not including <math|a> itself), then
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>\<leqslant\>lim<rsub|x\<rightarrow\>a>
g<around*|(|x|)>
</equation*>
provided both limits exist.\
</theorem>
\;
<\with|theorem-text|<macro|<localize|Theorem> (The Squeeze Theorem)>>
<\theorem*>
If <math|f<around*|(|x|)>\<leqslant\>g<around*|(|x|)>\<leqslant\>h<around*|(|x|)>>
when <math|x> is near <math|a> (not including <math|a> itself), and
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=lim<rsub|x\<rightarrow\>a>
h<around*|(|x|)>=L,
</equation*>
then
<\equation*>
lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>=L.
</equation*>
Note: the theorem also holds for one-sided limits.\
</theorem*>
</with>
<\wide-centered>
<image|<tuple|<#89504E470D0A1A0A0000000D494844520000024A000001840806000000C9290FC0000000097048597300000B1300000B1301009A9C180000200049444154789CECBD59901DC779E7FBCBCCAA3A4B9FD33B1A8D955848801BC4CDA4685143DB94A8906459638F2D2B14633FD87123E6C10F7238AEC3117EB0E3FAF1866F845F262C5FC97EB1F4A0EBB0C65A86A3A1258A16B550224570058895D81A8D46EFDBD9AA32F33E64559D3AA7BB41D06E4920953F44A1CEA9AA537B57FEEBFBBEFC3EF0783C1E8FC7E3F16C8AB0D6DA9FF74E783C1E8FC7E3F1DC8AC89FF70E783C1E8FC7E3F1DCAA78A1E4F1783C1E8FC7B3055E28793C1E8FC7E3F16C81174A1E8FC7E3F1783C5BE08592C7E3F1783C1ECF1678A1E4F1783C1E8FC7B3055E28793C1E8FC7E3F16C81174A1E8FC7E3F1783C5BF09E144AD65A3A9D4EFEBDDD6E638CA1D3E9B0BEBE8E31269FB7BEBE8EB59666B34992242449421CC768AD595F5FBFA9ED69ADF3ED5A6B31C610C731C6188C31B4DBEDED3D408FC7E3F1783C3F13DED399B933A1A294420841A7D3416B4DA552416B4DA3D1606060002925711C23A5444A992F57AD56DFD1F6E238260CC39EED5B6B514A6DF7A1793C1E8FC7E3F919F09EB428196358595941084110041863724B52A954CAC553B55ACDAD41524AD6D6D668B7DB944A25C230A4D96CBEA3ED8661C8F4F434AD560B802449BC35C9E3F1783C9E7731EF49A12484A052A92084C8BF974A25CAE5726E3502504AD1E97472AB4FB55ACDE7856148A552B9E96D6682E8E9A79FE6E4C993B900CBF6C1E3F1783C1ECFBB8FE0E7BD033F0D3271628CC985D1E5CB97397BF62C5353532C2C2C303434C4B163C778DFFBDE07905B929E79E619BEF18D6FB077EF5E7EF7777F97BD7BF7BEEDF6B2D82480AF7FFDEB5CBE7C999D3B77B27BF76E9224C9F7C3E3F1783C1ECFBB8BF7A450CA848BB596244972F75BBBDDE6CA952B7CFBDBDFE6F4E9D37CF4A31FE56FFFF66FE9743A7980F7F9F3E7F9C217BEC0C0C000070F1EBC29A104104511172E5CE095575EE1F2E5CBFCDAAFFD1ABB77EF464A4992244451F4D33C648FC7E3F1783C3F05DE9342298B4D9252D26EB7915272F0E0410E1E3CC8FBDFFF7EA4943CF3CC333CFDF4D35CBB768DE1E1610606060078F8E187F99DDFF91D5E7CF145DE7CF3CD9BDEA69492A79E7A8AE5E5652E5FBECCC9932779ECB1C7104278F79BC7E3F1783CEF52DE93FEA0627A8052A994F76A4B9284B1B1313EFCE10F73C71D77D0683478F1C517F3DF586BB9EFBEFBF8EC673FCB473EF2115657576F7A7B711CF3B5AF7D0D706EBCE3C78F333535958B368FC7E3F1783CEF3EB6512819C090E71AB0804E27DBDE65B0D9B2369BE296E959F6DF8F108272B99CBBDEACB53DDDFEF7EEDDCBC73EF63116161678EEB9E768B7DBF90050AFD7B9F3CE3B79F4D1476F6A7B4992303D3DCD2BAFBC8210828181014E9F3ECDC99327F3FDB911EFE10C0D1E8FC7E3F1BCABD926A164F2C16462C900094E2CD974C080D5A0356883B186188BCE4455B6EC3651142059AF3680E1E161FEF00FFF90DDBB77F32FFFF22F244942A552C997BF7CF932B3B3B37CF2939FEC492299ADB33880CB9FF4777FF77780EB2D57ABD53871E2043FF9C94F7AF6278EE30DFBD86EB769341ADB77D01E8FC7E3F178B68D6DB428D97CB0D99A03A0684CB199D9C859953279658B3FDF46368B0F9252522E97F35400E7CF9FE7FAF5EBB45AADDCEAD46C36A9D7EB3DEBC8924766DF8BE24908C18F7FFC639ACD26524AA22862666686575E7985A9A9A9DC5295A50C68B7DBB45A2D8C31846148B95CDEDE03F7783C1E8FC7B32D6C9F504A558FC2E99D968075056D55986F0558E9C493845CC308DC0FD5B6EE518FA8C9AC3F99701A1B1BE3631FFB184208CE9D3B97E75AFAE10F7FC8D4D4144F3EF964CFBA326154B42665EB3E77EE1CDFFBDEF7A8542A944A254AA512954A8513274EF0DDEF7E9752A90490A708C862968AE90B3211E5F1783C1E8FE7D6617B64892D0CC68D9AC00A4E3021E8B518A5D352BD84056209FAA750E9434AB9698C50AD56E3139FF804524ABEF5AD6F71E1C20500565656088280A3478FF6D48B2BE641CA4AA34829595A5AE2F39FFF7CEEBACBAC4CD56A959999195E78E1859E3829706E409F57C9E3F1783C9E5B9FED6BAD73A124C0420C348056717E511D09905854AA9EE274F869D8553673C14551C4FDF7DF8FD69A575E79852008387EFC38008F3DF618E0044D66892AD66BCBAC4900D3D3D33CF7DC73944A2594520441401445D4EB7556565678E9A59778EBADB772F75D1CC7586BF3F405D9FE79E1E4F1783C1ECFADC7F6B6CE16B0220F4B328521DF9C10580426353305E94E685CECF77685296DD6932CB3F864569FB1B1310E1D3AC4EAEA2A6B6B6BFCDBBFFD1B5A6B6EBFFD76DAED364AA9DCDD06F4944409828066B3C9ABAFBECAD5AB5731C6303030C0FAFA3A8B8B8B4829A9D56A5CBA7489A79E7ACA1D7D214D41E6722BBA053D1E8FC7E3F1DC5A6CBF194380B4500606810A693CB704A413491AD1931240DA6E3681EDE246E2A3D893EDF1C71F676565852F7CE10B08213870E000E07AAF65147BBC41D70D373D3DCD33CF3CC3CACA0AA55289A5A525DAED36711CB3B2B242A552A1D96CF295AF7C0570B1499995AAB87FC6181FA3E4F1783C1ECF2DC8F609252140A616170B35036316EA16842177B799542801DD14026910F876E6AFEE77B515454EE6466BB7DBECD9B3872B57AEF0CFFFFCCF3CFCF0C3DC7FFFFD349BCDFCF7C540EB62CF3770E54EBEFDED6F1345119393930C0C0C303939C9F8F838A3A3A354AB559ACD26172E5CE0273FF909C6184AA5524
</wide-centered>
<\example*>
<\equation*>
lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=?
</equation*>
For simplicity, first consider the case <math|x\<gtr\>0>. Then
<\equation*>
-x\<leqslant\> x*sin <frac|1|x>\<leqslant\>x,
</equation*>
and
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|+>>-x=lim<rsub|x\<rightarrow\>0<rsup|+>>
x=0.
</equation*>
By the squeeze theorem, we know
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|+>> x*sin <frac|1|x>=0.
</equation*>
Similarly, we have
<\equation*>
lim<rsub|x\<rightarrow\>0<rsup|->> x*sin <frac|1|x>=0.
</equation*>
So
<\equation*>
lim<rsub|x\<rightarrow\>0> x*sin <frac|1|x>=0.
</equation*>
</example*>
\;
<section|(skip)>
<section|Continuity>
<\definition>
A function <math|f> is said to be <strong|continuous> at <math|a>, if
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|x|)>=f<around*|(|a|)>.
</equation*>
If <math|f> is defined near <math|a> and <math|f> is not continuous at
<math|a>, we say <math|f> is <strong|discontinuous> at <math|a>.\
</definition>
\;
<\wide-block>
<tformat|<cwith|1|-1|1|-1|cell-lsep|2spc>|<cwith|1|-1|1|-1|cell-rsep|2spc>|<cwith|1|-1|1|-1|cell-bsep|2spc>|<cwith|1|-1|1|-1|cell-tsep|2spc>|<table|<row|<\cell>
<\example>
At which numbers is <math|f> continuous?
<image|<tuple|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
</example>
</cell>|<\cell>
<\itemize>
<item><math|f> is discontinuous at <math|1> since
<math|f<around*|(|1|)>> is undefined
<item><math|f> is discontinuous at <math|3> since
<math|lim<rsub|x\<rightarrow\>3> f<around*|(|x|)>> does not exist.\
<item><math|f> is discontinuous at <math|5> since
<math|lim<rsub|x\<rightarrow\>5>
f<around*|(|x|)>\<neq\>f<around*|(|5|)>>.\
<item><math|f> is continuous at other points.\
</itemize>
</cell>>>>
</wide-block>
\;
<\note>
A polynomial <math|P<around*|(|x|)>> is continuous for every
<math|x\<in\><with|font|Bbb|R>>.\
</note>
<\example>
Where are each of the following functions discontinuous?
<\enumerate-alpha>
<item><math|f<around|(|x|)>=<dfrac|x<rsup|2>-x-2|x-2>>
<\itemize>
<item>If <math|x\<neq\>2>, then <math|f> is continuous at <math|x.>
<item>If <math|x=2>, then <math|f> is discontinuous at <math|x>,
since <math|f<around*|(|2|)>> is undefined. However,\
<\equation*>
lim<rsub|x\<rightarrow\>2> f<around|(|x|)>=lim<rsub|x\<rightarrow\>2>
<dfrac|x<rsup|2>-x-2|x-2>=lim<rsub|x\<rightarrow\>2>
<dfrac|<around*|(|x+1|)>*<around*|(|x-2|)>|x-2>=lim<rsub|x\<rightarrow\>2>
<around*|(|x+1|)>=3.
</equation*>
If we define
<\equation*>
<wide|f|~><around*|(|x|)>=<choice|<tformat|<table|<row|<cell|<dfrac|x<rsup|2>-x-2|x-2>,>|<cell|x\<neq\>2>>|<row|<cell|3,>|<cell|x=2>>>>>
</equation*>
Then <math|f> is continuous everywhere (including <math|x=2>). Thus
this kind of discontinuity is called <strong|removable>.\
</itemize>
<item><math|f<around|(|x|)>=<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|2|2|cell-halign|l>|<cwith|1|-1|3|3|cell-halign|l>|<cwith|1|-1|3|3|cell-rborder|0ln>|<table|<row|<cell|<dfrac|1|x<rsup|2>>>|<cell|<text|if
>>|<cell|x\<neq\>0>>|<row|<cell|1>|<cell|<text|if
>>|<cell|x=0>>>>>|\<nobracket\>>>,\
<\equation*>
lim<rsub|x\<rightarrow\>0> f<around*|(|x|)>=\<infty\>
</equation*>
So the limit does not exist, and <math|f> can not be continuous at
<math|x=0>. This type of discontinuity is called <strong|infinite
discontinuity>.\
<item><math|f<around|(|x|)>=<around*|{|<tabular*|<tformat|<cwith|1|-1|1|1|cell-halign|l>|<cwith|1|-1|1|1|cell-lborder|0ln>|<cwith|1|-1|2|2|cell-halign|l>|<cwith|1|-1|3|3|cell-halign|l>|<cwith|1|-1|3|3|cell-rborder|0ln>|<table|<row|<cell|<dfrac|x<rsup|2>-x-2|x-2>>|<cell|<text|if
>>|<cell|x\<neq\>2>>|<row|<cell|1>|<cell|<text|if
>>|<cell|x=2>>>>>|\<nobracket\>>>
Similar to part (a), <math|f> is discontinuous at <math|x=2> since\
<\equation*>
lim<rsub|x\<rightarrow\>2> f<around*|(|x|)>=3\<neq\>f<around*|(|2|)>
</equation*>
This is still a removable discontinuity.\
<item><math|f<around*|(|x|)>=<around*|\<llbracket\>|x|\<rrbracket\>>>,
the greatest integer smaller than or equal to <math|x>. For example
<\equation*>
<around*|\<llbracket\>|1.3|\<rrbracket\>>=1,<space|1em><around*|\<llbracket\>|1|\<rrbracket\>>=1,<space|1em><around*|\<llbracket\>|-2.7|\<rrbracket\>>=-3.
</equation*>
<math|f> is continuous everywhere except when
<math|x\<in\>\<bbb-Z\>>: the set of all integers. The reason is
<\equation*>
lim<rsub|x\<rightarrow\>n<rsup|->>
f<around*|(|x|)>=n-1,<space|1em>lim<rsub|x\<rightarrow\>n<rsup|+>>
f<around*|(|x|)>=n<space|1em>\<Rightarrow\><space|1em>lim<rsub|x\<rightarrow\>n>
f<around*|(|x|)>*<text|<space|1em>does not exist>
</equation*>
where <math|n\<in\>\<bbb-Z\>>. So there is a jump at <math|x=n>. We
call it a <strong|jump discontinuity>.\
</enumerate-alpha>
\;
There are other cases of discontinuity. For example
<\equation*>
f<around*|(|x|)>=sin <frac|1|x>
</equation*>
<math|f<around*|(|x|)>> has no left or right hand limit as
<math|x\<rightarrow\>0>.\
</example>
<\definition>
A function <math|f<around*|(|x|)>> is said to be <strong|continuous
from the left> at <math|x=a> if
<\equation*>
lim<rsub|x\<rightarrow\>a<rsup|->> f<around*|(|x|)>=f<around*|(|a|)>.
</equation*>
Similarly we can define <strong|continuous from the right>.\
</definition>
\;
For example, the function <math|f<around*|(|x|)>=<around*|\<llbracket\>|x|\<rrbracket\>>>
is continuous from the right at <math|x=n\<in\>\<bbb-Z\>>.\
\;
<\definition>
A function <math|f<around*|(|x|)>> is said to be <strong|continuous on
an interval> <math|I> if it's continuous at every point
<math|x\<in\>I>. Here <math|I> can be all types of intervals, such as
<\equation*>
<around*|(|a,b|)>,<around*|[|a,b|]>,<around*|[|a,b|)>,<around*|[|a,\<infty\>|)>,<around*|(|-\<infty\>,a|)>*<text|<space|1em>etc.>
</equation*>
</definition>
\;
<\theorem>
If <math|f> and <math|g> are both continuous at <math|x=a>. Then
<\equation*>
f+g,f-g,f*g,c*f
</equation*>
are continuous at <math|x=a>, where <math|c> is a constant. Moreover,\
<\equation*>
<frac|f|g>
</equation*>
is continuous at <math|x=a> provided <math|g<around*|(|a|)>\<neq\>0>.
</theorem>
\;
<\theorem>
A polynomial <math|P<around*|(|x|)>> is continuous for all <math|x>. A
rational function <math|P<around*|(|x|)>/Q<around*|(|x|)>> is
continuous in its domain.\
</theorem>
\;
\;
<strong|Trigonometric functions>
From the geometrical definition of <math|sin x>, we know
<\equation*>
lim<rsub|x\<rightarrow\>0> sin x=0=sin 0
</equation*>
So <math|sin x> is continuous at <math|0>. Similarly we can show
<math|cos*x> is continuous at <math|0>.\
How about any <math|a>?
<\equation*>
sin <around*|(|a+x|)>=<around*|(|sin a|)>*<around*|(|cos
x|)>+<around*|(|cos a|)>*<around*|(|sin x|)>\<rightarrow\>sin
a<space|1em><text|as><space|1em>x\<rightarrow\>0
</equation*>
So <math|sin x> is continuous at every <math|a>. Similarly we can show
<math|cos x> is continuous at every <math|a>. From the quotient rule, \
<\equation*>
tan x=<frac|sin x|cos x>
</equation*>
is continuous in its domain, i.e. every <math|x> except
<math|x=<frac|2*k+1|2>*\<pi\>> for <math|k\<in\>\<bbb-Z\>>.\
<\question>
What type of discontinuity does <math|tan x> have at
\ <math|x=<frac|2*k+1|2>*\<pi\>> for <math|k\<in\>\<bbb-Z\>>?
</question>
Similarly, we deduce that all the trigonometric and inverse trigonometric
functions are continuous in their domains.\
\;
Other functions: root functions, exponential functions, logarithmic
functions are continuous at every number in the domains.\
<\theorem>
If <math|f> is continuous at <math|b> and
<math|lim<rsub|x\<rightarrow\>a> g<around*|(|x|)>=b>, then
<math|lim<rsub|x\<rightarrow\>a> f<around*|(|g<around*|(|x|)>|)>=f<around*|(|b|)>>,
i.e.\
<\equation*>
lim<rsub|x\<rightarrow\>a> f<around*|(|g<around*|(|x|)>|)>=f<around*|(|lim<rsub|x\<rightarrow\>a>
g<around*|(|x|)>|)>.
</equation*>
</theorem>
\;
<\theorem>
If <math|g> is continuous at <math|a> and <math|f> is continuous at
<math|g<around*|(|a|)>>, then <math|f\<circ\>g> is continuous at
<math|a>.\
</theorem>
\;
<\example>
The function
<\equation*>
f<around*|(|x|)>=sin <around*|(|1+x<rsup|2>|)>
</equation*>
is continuous at every number.\
</example>
<\example>
The function
<\equation*>
f<around*|(|x|)>=<frac|1+2<rsup|x>+sin x|<sqrt|1-cos x>>
</equation*>
is continuous everywhere except where <math|cos x=1>, i.e.
<math|x=2*k*\<pi\>> for <math|k\<in\>\<bbb-Z\>>.\
</example>
<\render-theorem|<strong|Intermediate Value Theorem>>
Suppose that <math|f> is continuous on the closed interval
<math|<around|[|a,b|]>> and let <math|m> be any number between
<math|f<around|(|a|)>> and <math|f<around|(|b|)>>, where
<math|f<around|(|a|)>\<neq\>f<around|(|b|)>>. Then there exists a
number <math|c> in <math|<around|(|a,b|)>> such that
<math|f<around|(|c|)>=m>.
</render-theorem>
\;
\;
Example. Show that the equation <math|x<rsup|2>-x-1=<frac|1|x+1>> has a
solution for <math|1\<less\>x\<less\>2>.
<strong|Answer:> Let
<\equation*>
f<around*|(|x|)>=x<rsup|2>-x-1-<frac|1|x+1>
</equation*>
We want to find <math|x> such that <math|f<around*|(|x|)>=0>. Now
<\equation*>
f<around*|(|1|)>\<less\>0,<space|1em>f<around*|(|2|)>\<gtr\>0
</equation*>
So <math|0> is a number between <math|f<around*|(|1|)>> and
<math|f<around*|(|2|)>>. Moreover, <math|f> is continuous in
<math|<around*|[|1,2|]>>. \ By the intermediate value theorem, there
exists an <math|x\<in\><around*|(|1,2|)>> such that
<math|f<around*|(|x|)>=0>.\
We can now pick the midpoint of <math|1> and <math|2>, that is
<math|<frac|3|2>>, then
<\equation*>
f<around*|(|<frac|3|2>|)>=<frac|9|4>-<frac|3|2>-1-<frac|2|5>\<less\>0.
</equation*>
Using the Intermediate value theorem again in
<math|<around*|[|<frac|3|2>,2|]>>, we know there exists an
<math|x\<in\><around*|(|<frac|3|2>,2|)>> such that
<math|f<around*|(|x|)>=0>.\
\;
Repeating this procedure, we can get more and more accurate
approximations of the root of <math|f<around*|(|x|)>>. This is called the
<strong|bisection method>.
<section|Limits at Infinity; Horizontal Asymptotes>
motivation: <math|lim<rsub|x\<rightarrow\>\<infty\>>
<frac|x<rsup|2>-1|x<rsup|2>+1>>
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> <frac|x<rsup|2>-1|x<rsup|2>+1>=lim<rsub|x\<rightarrow\>\<infty\>><frac|1-<frac|1|x<rsup|2>>|1+<frac|1|x<rsup|2>>>=<frac|1-0|1+0>=1
</equation*>
<\definition>
Suppose <math|f<around*|(|x|)>> is defined for <math|x\<gtr\>a> for
some <math|a>. Then we define
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=L
</equation*>
if <math|f<around*|(|x|)>> can be <strong|arbitrarily> close to
<math|L> as long as <math|x> is <strong|sufficiently> large. Similarly
we can define
<\equation*>
lim<rsub|x\<rightarrow\>-\<infty\>> f<around*|(|x|)>=L
</equation*>
</definition>
\;
<\definition>
If
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
f<around*|(|x|)>=L<space|1em>or<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
f<around*|(|x|)>=L,
</equation*>
we call <math|y=L> a <strong|horizontal asymptote> of <math|f>.
</definition>
\;
<\example>
Let <math|f<around*|(|x|)>=<frac|1|x>>. Then
<math|lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=0>. So
<math|y=0> is horizontal asymptote of <math|f>. Moreover,
<math|lim<rsub|x\<rightarrow\>0<rsup|+>> f<around*|(|x|)>=\<infty\>>,
so <math|x=0> is a vertical asymptote of <math|f>.
</example>
<\example>
\
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
<frac|3*x<rsup|2>-x-2|5*x<rsup|2>+4*x+1>=lim<rsub|x\<rightarrow\>\<infty\>>
<frac|3*-<frac|1|x>-<frac|2|x<rsup|2>>|5*+<frac|4|x>+<frac|1|x<rsup|2>>>=<frac|3|5>.
</equation*>
</example>
<\example>
\
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
<frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=lim<rsub|x\<rightarrow\>\<infty\>>
<frac|<sqrt|2*+<frac|1|x<rsup|2>>>|3**-<frac|5|x>>=<frac|<sqrt|2>|3>
</equation*>
<\equation*>
lim<rsub|x\<rightarrow\>-\<infty\>>
<frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=lim<rsub|x\<rightarrow\>-\<infty\>>
<frac|<frac|1|x><sqrt|2*x<rsup|2>+1>|3**-<frac|5|x>>=lim<rsub|x\<rightarrow\>-\<infty\>>
<frac|-<sqrt|2*+<frac|1|x<rsup|2>>>|3**-<frac|5|x>>=-<frac|<sqrt|2>|3>
</equation*>
So we have two horizontal asymptote <math|y=<sqrt|2>/3> and
<math|y=-<sqrt|2>/3>. Moreover
<\equation*>
lim<rsub|x\<rightarrow\><frac|5|3><rsup|+>>
<frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\><frac|5|3><rsup|->>
<frac|<sqrt|2*x<rsup|2>+1>|3*x*-5>=-\<infty\>.
</equation*>
So <math|x=5/3> is a vertical asymptote.
</example>
<unfolded|<\question>
<math|lim<rsub|x\<rightarrow\>\<infty\>> sin <frac|1|x>=?>
</question>|<\answer*>
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> sin
<frac|1|x>=lim<rsub|t\<rightarrow\>0<rsup|+>> sin t=sin 0=0.
</equation*>
</answer*>>
<unfolded|<\question>
<math|lim<rsub|x\<rightarrow\>\<infty\>> x*<around*|(|sin
<frac|1|x>|)>=?>
</question>|<\answer*>
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> x*<around*|(|sin
<frac|1|x>|)>=lim<rsub|x\<rightarrow\>\<infty\>> *<frac|sin
<frac|1|x>|<frac|1|x>>=lim<rsub|t\<rightarrow\>0<rsup|+>> <frac|sin
t|t>=1
</equation*>
</answer*>>
<\definition>
Suppose <math|f> is defined at every <math|x\<gtr\>a> for some
<math|a>. Then we define
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> f<around*|(|x|)>=\<infty\>
</equation*>
if <math|f<around*|(|x|)>> can be arbitrarily large as long as <math|x>
is sufficiently large. Similarly we can define
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
f<around*|(|x|)>=-\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
f<around*|(|x|)>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
f<around*|(|x|)>=-\<infty\>
</equation*>
</definition>
<\example>
\
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
x<rsup|2>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
x<rsup|2>=\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
x<rsup|3>=-\<infty\>.
</equation*>
</example>
<\note>
If <math|P<around*|(|x|)>> is a polynomial, then
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>>
P<around*|(|x|)>=\<pm\>\<infty\>,<space|1em>lim<rsub|x\<rightarrow\>-\<infty\>>
P<around*|(|x|)>=\<pm\>\<infty\>.
</equation*>
</note>
<\example>
A business manager determines that the total cost of producing <math|x>
units of a particular commodity may be modeled by the function
<\equation*>
C<around|(|x|)>=7.5*x+120,000
</equation*>
The average cost is <math|A<around|(|x|)>=<frac|C<around|(|x|)>|x>>.
Find <math|lim<rsub|x\<rightarrow\>+\<infty\>> A<around|(|x|)>> and
interpret your result.
</example>
<strong|Answer:>\
<\equation*>
lim<rsub|x\<rightarrow\>\<infty\>> <frac|7.5*x+120000|x>=7.5
</equation*>
<gap-wide|>
<section|Derivatives and Rates of Change>
tangent line, velocity, derivative, rates of change
\;
The slope of the secant line for a function <math|y=f<around*|(|x|)>>
from <math|x=a> to <math|x=b> is
<\equation*>
m=<frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>
</equation*>
If <math|b\<rightarrow\>a>, then the slope approaches the slope of the
tangent line if it exists at <math|x=a>.
<\definition>
If <math|f> is continuous at <math|a>, then we define
<\equation*>
f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
<frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>
</equation*>
to be the <strong|derivative> of <math|f> at <math|a>. If <math|f> has
a derivative at <math|a>, we say <math|f> is <strong|differentiable> at
<math|a>.
</definition>
\;
\;
Other motivations: velocity.\
Suppose <math|f<around*|(|t|)>> is the displacement of an object at time
<math|t>. Then the <strong|average velocity> from <math|t=a> to
<math|t=b> is
<\equation*>
<frac|f<around*|(|b|)>-f<around*|(|a|)>|b-a>.
</equation*>
The <strong|instantaneous velocity> at <math|t=a> is
<\equation*>
v=lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=f<rprime|'><around*|(|a|)>.
</equation*>
Similarly, the acceleration is the derivative of velocity:
<math|a=v<rprime|'>=f<rprime|''>>.
\;
Generally speaking, <math|y=f<rprime|'><around*|(|a|)>> is the
<strong|rate of change> of <math|y> with respect to <math|t> at <math|a>.
\;
<\example>
Find where the function <math|f<around*|(|x|)>=<around*|\||x|\|>> is
differentiable and find its derivative.
<\wide-centered>
<with|gr-mode|<tuple|edit|point>|gr-frame|<tuple|scale|1cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|0.4par|0.6par|center>|gr-grid|<tuple|empty>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|empty>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-color|blue|gr-auto-crop|true|<graphics||<with|magnify|0.6187192033860641|arrow-end|\<gtr\>|<line|<point|-0.6976773750669972|-0.4725822239854681>|<point|3.064795878819595|-0.4280933012526935>>>|<with|magnify|0.6187192033860641|arrow-end|\<gtr\>|<line|<point|1.1962703665956091|-1.0191569120991069>|<point|1.1390705004840198|2.0505631263079476>>>|<with|magnify|0.6187192033860641|color|blue|<line|<point|-0.4561665212172813|1.1798577574234392>|<point|1.1856707656828998|-0.4503127952722712>>>|<with|magnify|0.6187192033860641|color|blue|<line|<point|1.1856707785628005|-0.4503126636979932>|<point|2.5944867105132174|1.2751885515325685>>>|<with|color|blue|<math-at|<with|color|blue|y=<around*|\||x|\|>>|<point|1.6265770454169626|1.0483157106310128>>>|<with|color|blue|<text-at|corner|<point|0.36684727099403713|-0.7703082594865909>>>|<with|color|blue|<point|1.18567|-0.450313>>>>
</wide-centered>
<\enumerate>
<item>If <math|a\<gtr\>0>,
<\equation*>
lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|x\<rightarrow\>a>
<frac|<around*|\||x|\|>-<around*|\||a|\|>|x-a>=lim<rsub|x\<rightarrow\>a>
<frac|x-a|x-a>=1<space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|a|)>=1.
</equation*>
<item>If <math|a\<less\>0>,
<\equation*>
lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|x\<rightarrow\>a>
<frac|<around*|\||x|\|>-<around*|\||a|\|>|x-a>=lim<rsub|x\<rightarrow\>a>
<frac|-x+a|x-a>=-1<space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|a|)>=-1.
</equation*>
<item>If <math|a=0>,\
<\equation*>
lim<rsub|x\<rightarrow\>0> <frac|f<around*|(|x|)>-f<around*|(|0|)>|x>=lim<rsub|x\<rightarrow\>0>
<frac|<around*|\||x|\|>|x> <text| does not
exist><space|1em>\<Rightarrow\><space|1em>f<rprime|'><around*|(|0|)>
<text| does not exist.>
</equation*>
</enumerate>
</example>
Other examples where the function is not differentiable.
<\wide-centered>
<with|gr-mode|<tuple|edit|text-at>|gr-frame|<tuple|scale|0.75cm|<tuple|0.5gw|0.5gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|empty>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|empty>|gr-dash-style|11100|gr-color|red|gr-auto-crop|true|magnify|0.75|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|<graphics||<spline|<point|-5.4|0.1>|<point|-4.9|-0.7>|<point|-3.3|-0.2>>|<spline|<point|-3.3|0.8>|<point|-2.1000000000000005|1.6>|<point|-1.3000000000000003|1.5>>|<point|-3.3|-0.2>|<with|point-style|round|<point|-3.3|0.8>>|<spline|<point|0.3|0.5>|<point|1.8000000000000003|2.1>|<point|2.799999999999999|0.8>|<point|3.200000000000001|-1.0>|<point|3.8999999999999995|-0.8>>|<with|dash-style|11100|color|red|<line|<point|2.9|1.6>|<point|2.8999999999999995|-1.3>>>|<with|color|red|<point|2.9|-0.0339431>>|<with|color|red|<text-at|discontinuous|<point|-4|-1.2>>>|<with|color|red|<text-at|vertical
tangent|<point|1.8000000000000003|-1.8>>>>>
</wide-centered>
\;
\;
<\question>
Show that if <math|f<around*|(|a|)>> is defined and
<\equation*>
lim<rsub|x\<rightarrow\>a> <frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>
<text|exist>,
</equation*>
then <math|f> is continuous at <math|a>.
</question>
<\summarized-plain>
<section|The Derivative as a Function>
definition, graph, other notations
<|summarized-plain>
\;
</summarized-plain>
<\math>
<\equation*>
f<rprime|'><around*|(|a|)>=lim<rsub|x\<rightarrow\>a>
<frac|f<around*|(|x|)>-f<around*|(|a|)>|x-a>=lim<rsub|h\<rightarrow\>0>
<frac|f<around*|(|a+h|)>-f<around*|(|a|)>|h>
</equation*>
</math>
Now for a variable <math|x>, we obtain a function
<\equation*>
f<rprime|'><around*|(|x|)>=lim<rsub|h\<rightarrow\>0>
<frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>
</equation*>
called the <strong|derivative function> of <math|f>, or simply the
<strong|derivative> of <math|f>.
<\example>
Let <math|f<around*|(|x|)>=x<rsup|2>>, find
<math|f<rprime|'><around*|(|x|)>>.
\;
<strong|Answer:>
<\equation*>
lim<rsub|h\<rightarrow\>0> <frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
<frac|<around*|(|x+h|)><rsup|2>-x<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
<frac|2*x*h+h<rsup|2>|h>=lim<rsub|h\<rightarrow\>0>
<around*|(|2*x+h|)>=2*x
</equation*>
So
<\equation*>
f<rprime|'><around*|(|x|)>=2*x,<space|1em>x\<in\><with|font|Bbb|R>.
</equation*>
</example>
<\exercise>
What is <math|f<rprime|'><around*|(|x|)>> if
<math|f<around*|(|x|)>=x<rsup|n>,n\<in\>\<bbb-N\>>.
</exercise>
<\example>
Let <math|f<around*|(|x|)>=<around*|\||x|\|>>, find
<math|f<rprime|'><around*|(|x|)>>.
\;
<strong|Answer:>
<\equation*>
lim<rsub|h\<rightarrow\>0> <frac|f<around*|(|x+h|)>-f<around*|(|x|)>|h>=lim<rsub|h\<rightarrow\>0>
<frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>.
</equation*>
Consider three cases
<\enumerate>
<item><math|x\<gtr\>0>
<\equation*>
lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
<frac|x+h-x|h>=1.
</equation*>
<item><math|x\<less\>0>
<\equation*>
lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
<frac|-<around*|(|x+h|)>+x|h>=-1.
</equation*>
<item><math|x=0.>
<\equation*>
lim<rsub|h\<rightarrow\>0> <frac|<around*|\||x+h|\|>-<around*|\||x|\|>|h>=lim<rsub|h\<rightarrow\>0>
<frac|<around*|\||h|\|>|h> <text|does not exist>.
</equation*>
</enumerate>
So
<\equation*>
f<rprime|'><around*|(|x|)>=<choice|<tformat|<table|<row|<cell|1,>|<cell|x\<gtr\>0,>>|<row|<cell|-1,>|<cell|x\<less\>0,>>|<row|<cell|<text|undefined,>>|<cell|x=0.>>>>>
</equation*>
</example>
Given the graph of <math|f>, we can sketch the graph of
<math|f<rprime|'>>.
\;
<\example>
\;
\;
<\wide-tabular>
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</wide-tabular>
</example>
<\render-theorem>
Other notations for derivative
<|render-theorem>
For a function <math|y=f<around*|(|x|)>>, the derivative function is
<\equation*>
f<rprime|'><around*|(|x|)>,<space|1em>y<rprime|'>,<space|1em><frac|d*y|d*x>,<space|1em>D<rsub|x>
f,<space|1em>D f<space|1em>
</equation*>
The derivative at <math|a> is
<\equation*>
f<rprime|'><around*|(|a|)>,<space|1em><space|1em><around*|\<nobracket\>|<frac|d*y|d*x>|\|><rsub|x=a>,<space|1em>D<rsub|x>
f<around*|(|a|)>,<space|1em>D f<around*|(|a|)>.
</equation*>
</render-theorem>
\;
\;
\;
\;
\;
\;
\;
</shown>>
</body>
<\initial>
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<associate|eqn-ver-sep|<macro|0fn>>
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<associate|auto-10|<tuple|4|7>>
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<associate|auto-12|<tuple|6|10>>
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<associate|auto-7|<tuple|2.2|3>>
<associate|auto-8|<tuple|2.3|4>>
<associate|auto-9|<tuple|3|5>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|toc>
<vspace*|2fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|font-size|<quote|1.19>|Chapter
2: Limits and Derivatives> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|1fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>The
Tangent and Velocity Problems> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.5fn>
<with|par-left|<quote|1tab>|1.1<space|2spc>The tangent problem (how to
find the tangent line) <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3>>
<with|par-left|<quote|1tab>|1.2<space|2spc>The velocity problem (how to
find instantaneous velocity) <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-4>>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
Limit of a Function> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-5><vspace|0.5fn>
<with|par-left|<quote|1tab>|2.1<space|2spc>Intuitive definition of a
limit <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6>>
<with|par-left|<quote|1tab>|2.2<space|2spc>One-sided limits
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7>>
<with|par-left|<quote|1tab>|2.3<space|2spc>Infinite limits
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-8>>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Calculating
Limits Using the Limit Laws> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-9><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>(skip)>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-10><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>Continuity>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-11><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|6<space|2spc>Limits
at Infinity; Horizontal Asymptotes>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-12><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|7<space|2spc>Derivatives
and Rates of Change> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-13><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|8<space|2spc>The
Derivative as a Function> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-14><vspace|0.5fn>
</associate>
</collection>
</auxiliary>