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The concept of limits is quite clear to me. I used Calculus Early Transcendentals by James Stewart for learning the basic concepts of limits and even solved the problems given in it but on the internet and in some exams we have some difficult problems of evaluation of limits. I can cite an example problem to illustrate myself more properly $$ \lim_{x \to 0}~~ [tan ~~({\pi/4 + x})]^{1/x}$$.

These questions involve some knowledge which is beyond my current knowledge of limits. I searched the internet and found that textbooks are meant and focuses more epsilon-delta definition and it's proofs and questions like these are just for competitive exams and therefore I was unable to find these limits laws (derived limit laws for solving problems) in well known books like Spivak Calculus, Thomas Calculus, James Stewart Calculus etc.

There is a thread on MSE where @DaveL.Renfro shares his lecture notes and it's just great, I want something like that (more information than that). The reasoning he has used was also very agreeable, so I request you all to please refer me resources as per my need.

Thank you. Any suggestion might be helpful.

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  • $\begingroup$ Poor generation who was forced to pay for Stewart's Integral House and got in return that horrible book that teaches nothing. What are the references in the link missing that you still need? $\endgroup$ Nov 7, 2019 at 12:03
  • $\begingroup$ @conditionalMethod What is it that your indicating at? I really can't understand your comment on Stewart. $\endgroup$ Nov 7, 2019 at 12:10
  • $\begingroup$ @conditionalMethod The missing things are limits of a series, Newton's Leibniz Method and limit of fractional part function. $\endgroup$ Nov 7, 2019 at 12:12
  • $\begingroup$ He (and publishers) sold those books supper expensive, huge books, and one of the worse books in Calculus that I have ever seen used in universities. With the money he earned he build a fancy custom house in Toronto that people call Integral house. Probably the only good thing that those books produced. $\endgroup$ Nov 7, 2019 at 12:12
  • $\begingroup$ @conditionalMethod It may be worse because your country have some very good books. If you were to study in India then I'm not sure that you will retain on your view on Stewart Calculus. $\endgroup$ Nov 7, 2019 at 12:14

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I think I might have a solution for that question in particular:

$$L = \lim_{x\rightarrow0}[\tan(\frac{\pi}{4} + x)]^\frac{1}{x}$$

Let $a=\frac{1}{x}$

As $x\rightarrow 0$, $a\rightarrow\infty$

$$L = \lim_{a\rightarrow\infty}[\tan(\frac{\pi}{4} + \frac{1}{a})]^a$$ $$L = [\lim_{a\rightarrow\infty}\tan(\frac{\pi}{4} + \frac{1}{a})]^{\lim_{a\rightarrow\infty}a}$$ $$L = [\tan(\frac{\pi}{4} + 0)]^{\lim_{a\rightarrow\infty}a}$$ $$L = [1]^{\lim_{a\rightarrow\infty}a}$$ $$L = 1$$

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  • $\begingroup$ The limit is actually $e^2$. $\endgroup$ Oct 30, 2021 at 4:56
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I don't have any resource recommendations at hand, but I can give you a tip for dealing with limits of the form $\lim_{x\to a}f(x)^{g(x)}$: rewrite $f(x)^{g(x)}$ as $e^{g(x)\ln(f(x))}$. Because of the continuity of the exponential function, this will boil down the problem of evaluating $\lim_{x\to a}f(x)^{g(x)}$ to that of evaluating

$$\lim_{x\to a}g(x)\ln(f(x))$$

In the case of evaluating $\lim_{x\to 0}\left(\tan\left(\frac{\pi}{4}+x\right)\right)^{\frac{1}{x}}$ (the limit is $e^2$, as will be seen), it would be enough to compute

$$\lim_{x\to 0}\frac{\ln\left(\tan\left(\frac{\pi}{4}+x\right)\right)}{x}$$

Note that numerator and denominator both approach $0$ as $x\to 0$. Since

$$\lim_{x\to 0}\frac{\frac{d}{dx}\ln\left(\tan\left(\frac{\pi}{4}+x\right)\right)}{\frac{d}{dx}x}=\lim_{x\to 0}\frac{\frac{\sec^2\left(\frac{\pi}{4}+x\right)}{\tan\left(\frac{\pi}{4}+x\right)}}{1}=\frac{\sec^2\left(\frac{\pi}{4}+0\right)}{\tan\left(\frac{\pi}{4}+0\right)}=\frac{1}{\cos^2\left(\frac{\pi}{4}\right)}=2$$

it follows from L'Hôpital's Rule that

$$\lim_{x\to 0}\frac{\ln\left(\tan\left(\frac{\pi}{4}+x\right)\right)}{x}=2$$

Thus,

$$\lim_{x\to 0}e^{\frac{1}{x}\ln\left(\tan\left(\frac{\pi}{4}+x\right)\right)}=e^2$$

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