> <\body> <\hide-preamble> >>>>> \; |gr-frame|>|gr-geometry||gr-grid||1>|gr-grid-old||1>|gr-edit-grid-aspect|||>|gr-edit-grid||1>|gr-edit-grid-old||1>|gr-fill-color|red|gr-color|blue|||>||>|>|>>||)>|>>||)>|>>||>>||>>|-f|b-a>*+f|>>| \; >>| In fact, <\equation*> m=lima> -f|b-a> >>| tangent line: <\equation*> y=m*x+f \ for some slope m >>|<\with|fill-color|red|color|blue> \; >> <\unfolded-plain> Let > be the position of an object at time . The average velocity from to is <\equation*> -S|b-a> The instantaneous velocity is the limit of the average velocity in shorter and shorter time intervals, i.e. <\equation*> lima> -S|b-a> <\remark> Limit is always a dynamic process! <|unfolded-plain> \; <\definition> For a given function >, and a point >, we say <\equation*> a> f=L>>, read as: the limit of >, as approaches , equals , if > gets arbitrarily close to as long as is sufficiently close to . <\example> Let =x>, then\ <\equation*> lim3> f=9 <\wide-block> |||| <\example> Find the equation of the tangent line to the graph of =x> at . <\answer*> Draw a secant line from > to |)>>. Its slope is <\equation*> m=-1|x-1>=-1|x-1>. Then the slope of the tangent line is <\equation*> lim1> m=lim1> -1|x-1>=lim1> x+1=2. So the equation of the tangent line is <\equation*> y=2*+1=2*x-1. >>> <\question> \; <\equation*> lim0> =? Hint: draw a circle. |gr-frame|>|gr-geometry||gr-grid||2>|gr-grid-old||2>|gr-edit-grid-aspect|||>|gr-edit-grid||2>|gr-edit-grid-old||2>|gr-snap||gr-arrow-end|\|gr-color|blue|||>||>>||>>|>>|>>||>>|||>>|>>|||>>|>|0> =1|>>>> <\example> \; <\equation*> lim0> sin =? The limit does not exist!\ <\example> \; <\equation*> lim0> x*sin =0 In the definition of a> f=L>, the function value > is irrelevant. In fact, can even be undefined on . > <\example> Let <\equation*> f=|0,>>|,>|0,>>||>>>>g=|0,>>|,>|0.>>>>> Find 0> f> and 0> g>. <\answer*> <\equation*> lim0> f=1\f <\equation*> lim0> g=1,g <\definition> For a given function >, and a point , \ we say <\equation*> >> f=L>>, read as: the of >, as approaches , equals , if > gets arbitrarily close to as long as a> is sufficiently close to . Definition of the is similar.\ <\example> Find the left and right hand limits of <\equation*> f=|0,>>||0.>>>>> <\answer*> <\equation*> lim0> f=-1,lim0> f=3. \; Relation between limit and one-sided limits: <\itemize> If the limit exists, the both left and right hand limit exsits and equal to the limit value itself If both left and right hand limit exists and equal to the same value, then the limit exists and equal to the same value. If both left and right hand limit exists but they are not equal, then the limit does not exist.\ \; <\example> \ <\wide-tabular> || |png>|0.5par|||> |<\cell> <\itemize> 2> g=3>> 2> g=1>> 2> g>> not exist! > is undefined! 5> g=2>> 5> g=2>> 5> g=2>> \1.2> 3> g=1.5=g>> >>> <\definition> We write <\equation*> lima> f=\ if > gets arbitrarily large when is sufficiently close to . Similarly, we can define <\equation*> lima> f=-\ <\note*> In these cases we will not say the limit exists.\ <\example*> <\equation*> lim0> >=\ can be defined similarly <\equation*> lima> f=\-\ <\equation*> lima> f=\-\ <\example*> <\equation*> lim0> =\,lim0> =-\. <\example*> \; <\equation*> lim0> ln x=-\,lim|2>|)>> tan x=\,lim|2>|)>> tan x=-\ <\definition> If the left/right hand limit of > is > or > as approaches , then the line is called a vertical asymptote of .\ \; <\wide-block> |||| (for limit or one-sided limits) <\enumerate> Sum Law: a> +g|]>=lima> f+lima> g>> provided the limits on the right-hand-side exist.\ Difference Law: a> -g|]>=lima> f-lima> g>> provided the limits on the right-hand-side exist.\ Constant Multiple Law: a> c*f=c*lima>> f> provided the limit on the right-hand-side exists, where is a constant. \ Product Law: a> *g|]>=lima> f*lima> g>> provided the limits on the right-hand-side exist.\ Quotient Law: \ a> |g>=a> f>|a> g>>>> provided the limits on the right-hand-side exist and a> g\0>.\ >>> \; Generalize to any of functions: e.g.\ <\equation*> lima> 2*f+5*g=2*lima> f+5*lima> g. Power Law: a> f=a> f|]>>.\ Root Law: a> f=a> f|]>> ( must be odd if a> f\0>).\ Two simple limits: <\equation*> lima> c=c,lima> x=a. \; <\wide-block> |||| for polynomials and rational functions: <\equation*> lima> P=P,lima> |Q>=|Q>,Q\0. where are polynomials.\ >>> \; <\wide-block> |||| <\example> \; <\equation*> lim2> -4x+5|)>=2-4\2+5=5. <\equation*> lim1> -2x|x+1>==-. <\equation*> lim1> -2x|x-1>=lim1> |+x+1|)>> does not exist> In fact <\equation*> lim1> |+x+1|)>>=\,lim1> |+x+1|)>>=-\ Another example: <\equation*> lim1> -x|x-1>=lim1>|+x+1|)>>=lim1>+x+1|)>>=. >>> \; <\wide-block> |||| If =g> when a>, then a> f=lima> g>, provided the limits exist. >>> \; <\theorem> If \g> when is near (not including itself), then <\equation*> lima> f\lima> g provided both limits exist.\ \; <\with|theorem-text| (The Squeeze Theorem)>> <\theorem*> If \g\h> when is near (not including itself), and <\equation*> lima> f=lima> h=L, then <\equation*> lima> g=L. Note: the theorem also holds for one-sided limits.\ <\wide-centered> |png>|0.3par|||> <\example*> <\equation*> lim0> x*sin =? For simplicity, first consider the case 0>. Then <\equation*> -x\ x*sin \x, and <\equation*> lim0>-x=lim0> x=0. By the squeeze theorem, we know <\equation*> lim0> x*sin =0. Similarly, we have <\equation*> lim0> x*sin =0. So <\equation*> lim0> x*sin =0. \; <\definition> A function is said to be at , if <\equation*> lima> f=f. If is defined near and is not continuous at , we say is at .\ \; <\wide-block> |||| <\example> At which numbers is continuous? |png>|0.4par|||> |<\cell> <\itemize> is discontinuous at since > is undefined is discontinuous at since 3> f> does not exist.\ is discontinuous at since 5> f\f>.\ is continuous at other points.\ >>> \; <\note> A polynomial > is continuous for every >.\ <\example> Where are each of the following functions discontinuous? <\enumerate-alpha> =-x-2|x-2>> <\itemize> If 2>, then is continuous at If , then is discontinuous at , since > is undefined. However,\ <\equation*> lim2> f=lim2> -x-2|x-2>=lim2> *|x-2>=lim2> =3. If we define <\equation*> =-x-2|x-2>,>|2>>||>>>> Then is continuous everywhere (including ). Thus this kind of discontinuity is called .\ =|||||>>|>|0>>||>|>>>>|\>>,\ <\equation*> lim0> f=\ So the limit does not exist, and can not be continuous at . This type of discontinuity is called .\ =|||||-x-2|x-2>>|>|2>>||>|>>>>|\>> Similar to part (a), is discontinuous at since\ <\equation*> lim2> f=3\f This is still a removable discontinuity.\ =|x|\>>, the greatest integer smaller than or equal to . For example <\equation*> |1.3|\>=1,|1|\>=1,|-2.7|\>=-3. is continuous everywhere except when \>: the set of all integers. The reason is <\equation*> limn> f=n-1,limn> f=n\limn> f*does not exist> where \>. So there is a jump at . We call it a .\ \; There are other cases of discontinuity. For example <\equation*> f=sin > has no left or right hand limit as 0>.\ <\definition> A function > is said to be at if <\equation*> lima> f=f. Similarly we can define .\ \; For example, the function =|x|\>> is continuous from the right at \>.\ \; <\definition> A function > is said to be if it's continuous at every point I>. Here can be all types of intervals, such as <\equation*> ,,,|)>,,a|)>*etc.> \; <\theorem> If and are both continuous at . Then <\equation*> f+g,f-g,f*g,c*f are continuous at , where is a constant. Moreover,\ <\equation*> is continuous at provided \0>. \; <\theorem> A polynomial > is continuous for all . A rational function /Q> is continuous in its domain.\ \; \; From the geometrical definition of , we know <\equation*> lim0> sin x=0=sin 0 So is continuous at . Similarly we can show is continuous at .\ How about any ? <\equation*> sin =*+*\sin ax\0 So is continuous at every . Similarly we can show is continuous at every . From the quotient rule, \ <\equation*> tan x= is continuous in its domain, i.e. every except *\> for \>.\ <\question> What type of discontinuity does have at \ *\> for \>? 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 is continuous at and a> g=b>, then a> f|)>=f>, i.e.\ <\equation*> lima> f|)>=fa> g|)>. \; <\theorem> If is continuous at and is continuous at >, then g> is continuous at .\ \; <\example> The function <\equation*> f=sin |)> is continuous at every number.\ <\example> The function <\equation*> f=+sin x|> is continuous everywhere except where , i.e. > for \>.\ <\render-theorem|> Suppose that is continuous on the closed interval > and let be any number between > and >, where \f>. Then there exists a number in > such that =m>. \; \; Example. Show that the equation -x-1=> has a solution for x\2>. Let <\equation*> f=x-x-1- We want to find such that =0>. Now <\equation*> f\0,f\0 So is a number between > and >. Moreover, is continuous in >. \ By the intermediate value theorem, there exists an > such that =0>.\ We can now pick the midpoint of and , that is >, then <\equation*> f|)>=--1-\0. Using the Intermediate value theorem again in ,2|]>>, we know there exists an ,2|)>> such that =0>.\ \; Repeating this procedure, we can get more and more accurate approximations of the root of >. This is called the . motivation: \> -1|x+1>> <\equation*> lim\> -1|x+1>=lim\>>|1+>>==1 <\definition> Suppose > is defined for a> for some . Then we define <\equation*> lim\> f=L if > can be close to as long as is large. Similarly we can define <\equation*> lim-\> f=L \; <\definition> If <\equation*> lim\> f=Lorlim-\> f=L, we call a of . \; <\example> Let =>. Then \> f=0>. So is horizontal asymptote of . Moreover, 0> f=\>, so is a vertical asymptote of . <\example> \ <\equation*> lim\> -x-2|5*x+4*x+1>=lim\> ->|5*++>>=. <\example> \ <\equation*> lim\> +1>|3*x*-5>=lim\> >>|3**->=|3> <\equation*> lim-\> +1>|3*x*-5>=lim-\> +1>|3**->=lim-\> >>|3**->=-|3> So we have two horizontal asymptote /3> and /3>. Moreover <\equation*> lim> +1>|3*x*-5>=\,lim> +1>|3*x*-5>=-\. So is a vertical asymptote. \> sin =?> |<\answer*> <\equation*> lim\> sin =lim0> sin t=sin 0=0. > \> x*|)>=?> |<\answer*> <\equation*> lim\> x*|)>=lim\> *|>=lim0> =1 > <\definition> Suppose is defined at every a> for some . Then we define <\equation*> lim\> f=\ if > can be arbitrarily large as long as is sufficiently large. Similarly we can define <\equation*> lim\> f=-\,lim-\> f=\,lim-\> f=-\ <\example> \ <\equation*> lim\> x=\,lim-\> x=\,lim-\> x=-\. <\note> If > is a polynomial, then <\equation*> lim\> P=\\,lim-\> P=\\. <\example> A business manager determines that the total cost of producing units of a particular commodity may be modeled by the function <\equation*> C=7.5*x+120,000 The average cost is =|x>>. Find +\> A> and interpret your result. \ <\equation*> lim\> =7.5 tangent line, velocity, derivative, rates of change \; The slope of the secant line for a function > from to is <\equation*> m=-f|b-a> If a>, then the slope approaches the slope of the tangent line if it exists at . <\definition> If is continuous at , then we define <\equation*> f=lima> -f|x-a> to be the of at . If has a derivative at , we say is at . \; \; Other motivations: velocity.\ Suppose > is the displacement of an object at time . Then the from to is <\equation*> -f|b-a>. The at is <\equation*> v=lima> -f|x-a>=f. Similarly, the acceleration is the derivative of velocity: =f>. \; Generally speaking, > is the of with respect to at . \; <\example> Find where the function => is differentiable and find its derivative. <\wide-centered> |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-grid-old||1>|gr-edit-grid-old||1>|gr-color|blue|gr-auto-crop|true|||>>|||>>||>>||>>|>|>>|>>|>>> <\enumerate> If 0>, <\equation*> lima> -f|x-a>=lima> -|x-a>=lima> =1\f=1. If 0>, <\equation*> lima> -f|x-a>=lima> -|x-a>=lima> =-1\f=-1. If ,\ <\equation*> lim0> -f|x>=lim0> |x> \f Other examples where the function is not differentiable. <\wide-centered> |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-dash-style|11100|gr-color|red|gr-auto-crop|true|magnify|0.75|gr-grid-old||1>|gr-edit-grid-old||1>|||>|||>||>|||||>||>>|>|>>|>>>> \; \; <\question> Show that if > is defined and <\equation*> lima> -f|x-a> , then is continuous at . <\summarized-plain> definition, graph, other notations <|summarized-plain> \; <\math> <\equation*> f=lima> -f|x-a>=lim0> -f|h> Now for a variable , we obtain a function <\equation*> f=lim0> -f|h> called the of , or simply the of . <\example> Let =x>, find >. \; <\equation*> lim0> -f|h>=lim0> -x|h>=lim0> |h>=lim0> =2*x So <\equation*> f=2*x,x\. <\exercise> What is > if =x,n\\>. <\example> Let =>, find >. \; <\equation*> lim0> -f|h>=lim0> -|h>. Consider three cases <\enumerate> 0> <\equation*> lim0> -|h>=lim0> =1. 0> <\equation*> lim0> -|h>=lim0> +x|h>=-1. <\equation*> lim0> -|h>=lim0> |h> . So <\equation*> f=|0,>>||0,>>|>|>>>> Given the graph of , we can sketch the graph of >. \; <\example> \; \; <\wide-tabular> | |gr-frame|>|gr-geometry||gr-grid||gr-edit-grid-aspect|||>|gr-edit-grid||gr-grid-aspect-props|||>|gr-grid-aspect||>|gr-edit-grid-old||1>|gr-color|red|gr-auto-crop|true|gr-grid-old||1>||>|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|>|>|>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||>>|>|>>>|>|>>>>>>> >>> <\render-theorem> Other notations for derivative <|render-theorem> For a function >, the derivative function is <\equation*> f,y,,D f,D f The derivative at is <\equation*> f,||\|>,D f,D f. \; \; \; \; \; \; \; > <\initial> <\collection> > > > > <\references> <\collection> > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||font-size||Chapter 2: Limits and Derivatives> |.>>>>|> |math-font-series||1The Tangent and Velocity Problems> |.>>>>|> |1.1The tangent problem (how to find the tangent line) |.>>>>|> > |1.2The velocity problem (how to find instantaneous velocity) |.>>>>|> > |math-font-series||2The Limit of a Function> |.>>>>|> |2.1Intuitive definition of a limit |.>>>>|> > |2.2One-sided limits |.>>>>|> > |2.3Infinite limits |.>>>>|> > |math-font-series||3Calculating Limits Using the Limit Laws> |.>>>>|> |math-font-series||4(skip)> |.>>>>|> |math-font-series||5Continuity> |.>>>>|> |math-font-series||6Limits at Infinity; Horizontal Asymptotes> |.>>>>|> |math-font-series||7Derivatives and Rates of Change> |.>>>>|> |math-font-series||8The Derivative as a Function> |.>>>>|>