#119 part a 2n +5:31llRealAnalysis-ISBN-fix...3. cosr.4. The other trig functions: tan(z), cot(z), sec(r), and ese(r).Problem 118. What allows us to conclude that S(z) = sin (e") is continuousat any point a without referring back to the definition of continuity?Theorem (I can also be used to study the convergence of sequences. Forexample, since f(z) - e is contimuous at any point and lim, - 1,then lim e )-e. This also illustrates a certain way of thinking aboutcontimuous functions. They are the ones where we can "commute" the functionand a limit of a sequence. Specifically, if f is continIous at a and lim, In =a,then lim, () - f(a) - f (lim Fn).Problem 119. Compute the following limits. Be sure to point out how conti-Ruity is inroleed.(a) lim sin )S 2n +(b) lim V +1(e) lim eln (1/)Having this rigorous formulation of continuity is necessary for proving theExtreme Value Theorem and the Mean Value Theorem. However there is onemore piece of the puzzle to address before we can prove these theorems.We will do this in the next chapter, but before we go on it is time to define afundamental concept that was probably one of the first you learned in caleulus:limits.6.3The Definition of the Limit of a FunctionSince these days the limit concept is generally regarded as the starting point forcalculus, you might think it is a little strange that we've chosen to talk aboutCONTINUITY: WHAT IT IsN'T AND WHAT IT Is116contimuity first. But historically, the formal definition of a limit came afterthe formal definition of continmity. In some ways, the limit concept was partof a unification of all the ideas of calculus that were studied previously and,subsequently, it became the basis for all ideas in calculus. For this reason it islogical to make it the first topie covered in a caleulus course.To be sure, limits were always lurking in the background. In his attempts tojustify his caleulations, Newton used what he called his doctrine of "UltimateRatios." Specifically the ratio tA- - ehtA- 2r +h becomes theultimate ratio 2z at the last instant of time before h - an "evanescent quantity"- vanishes (. p. 33). Similarly Leibniz's "infinitely small" differentials dr anddy can be seen as an attempt to get "arbitrarily close" to z and y, respectively.This is the idea at the heart of caleulus: to get arbitrarily close to, say, r withoutactually reaching it.As we saw in Chapter Lagrange tried to avoid the entire issue of "arbitraryclosesness," both in the limit and differential forms when, in 1797, he attemptedto found calculus on infinite series.Although Lagrange's efforts failed, they set the stage for Cauchy to providea definition of derivative which in turn relied on his precise formulation of alimit. Consider the following example: to determine the slope of the tangentline (derivative) of f(z) = sin z at z=0. We consider the graph of the differencequotient D(x) -Next1DashboardCalendarTo DoNotificationsInbox

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#119 part a 2n +5:31llRealAnalysis-ISBN-fix...3. cosr.4. The other trig functions: tan(z), cot(z), sec(r), and ese(r).Problem 118. What allows us to conclude that S(z) = sin (e&#34;) is continuousat any point a without referring back to the definition of continuity?Theorem (I can also be used to study the convergence of sequences. Forexample, since f(z) - e is contimuous at any point and lim, - 1,then lim e )-e. This also illustrates a certain way of thinking aboutcontimuous functions. They are the ones where we can &#34;commute&#34; the functionand a limit of a sequence. Specifically, if f is continIous at a and lim, In =a,then lim, () - f(a) - f (lim Fn).Problem 119. Compute the following limits. Be sure to point out how conti-Ruity is inroleed.(a) lim sin )S 2n +(b) lim V +1(e) lim eln (1/)Having this rigorous formulation of continuity is necessary for proving theExtreme Value Theorem and the Mean Value Theorem. However there is onemore piece of the puzzle to address before we can prove these theorems.We will do this in the next chapter, but before we go on it is time to define afundamental concept that was probably one of the first you learned in caleulus:limits.6.3The Definition of the Limit of a FunctionSince these days the limit concept is generally regarded as the starting point forcalculus, you might think it is a little strange that we've chosen to talk aboutCONTINUITY: WHAT IT IsN'T AND WHAT IT Is116contimuity first. But historically, the formal definition of a limit came afterthe formal definition of continmity. In some ways, the limit concept was partof a unification of all the ideas of calculus that were studied previously and,subsequently, it became the basis for all ideas in calculus. For this reason it islogical to make it the first topie covered in a caleulus course.To be sure, limits were always lurking in the background. In his attempts tojustify his caleulations, Newton used what he called his doctrine of &#34;UltimateRatios.&#34; Specifically the ratio tA- - ehtA- 2r +h becomes theultimate ratio 2z at the last instant of time before h - an &#34;evanescent quantity&#34;- vanishes (. p. 33). Similarly Leibniz's &#34;infinitely small&#34; differentials dr anddy can be seen as an attempt to get &#34;arbitrarily close&#34; to z and y, respectively.This is the idea at the heart of caleulus: to get arbitrarily close to, say, r withoutactually reaching it.As we saw in Chapter Lagrange tried to avoid the entire issue of &#34;arbitraryclosesness,&#34; both in the limit and differential forms when, in 1797, he attemptedto found calculus on infinite series.Although Lagrange's efforts failed, they set the stage for Cauchy to providea definition of derivative which in turn relied on his precise formulation of alimit. Consider the following example: to determine the slope of the tangentline (derivative) of f(z) = sin z at z=0. We consider the graph of the differencequotient D(x) -Next1DashboardCalendarTo DoNotificationsInbox

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Answered: Image /qna-images/answer/0d4676d3-8191-46ee-8d2c-42154beac2ca.jpg

Problem 119. Compute the following limits. Be sure to point out how conti- nuity is inroleed (a) lim sin 2n