I need some resources for learning the evaluation of limits.

Aug 2019
88
23
India
The concepts of limits are quite clear to me, I did it from James Stewart Calculus and concepts are quite concrete too. But when it comes to evaluating the limits, I just can't figure out how to proceed, I mean there are some rules like we always try to do something so that zero doesn't ever occur in denominator, some little approximations but standard books don't focus on techniques of finding the limits, I have gone for many famous books like Spivak Calculus, Thomas Calculus etc. and found that they focus on differential and integral calculus much (well they are very right too in doing so).

Competitive exams, online problems or University exams contain some weird limit questions which requires greater knowledge than what these books contain. These weird limit questions need concepts like:- L'Hôpitals rule, Newton Leibniz rule, Greatest integer function, fractional part functionetc. Here is a link https://math.stackexchange.com/questions/2760914/mathematics-books-to-master-limits where Dave L. Renfro shares his personal lecture notes and it contains some very great knowledge, his reasoning is very much agreeable but they are not complete.

So I request that can someone please cite me some references for my above said problem? Can anyone please share their personal notes on limits?
 
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Dec 2015
1,084
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Earth
This degree ? For example \(\displaystyle l=\displaystyle \lim_{n\rightarrow \infty } \frac{1^n +2^n + 3^n +...+n^n }{(n!)^2}\).

Solution : \begin{align*}
(n!)^2 &= \underbrace{1 \times 1 \times 2 \times 2 \times \dots \times \left(\frac{n}{2} -1\right) \times \left(\frac{n}{2} -1\right)}_{\geq ~ 1 \times 1 \times 2 \times 2 \times \dots \times 2 \times 2 \\ \qquad= 2^{n-4}} \times \underbrace{\frac{n}{2} \times \frac{n}{2} \dots (n-1) \times (n-1) \times n \times n}_{\geq ~ \frac{n}{2} \times \frac{n}{2} \times \dots \times \frac{n}{2} \times \frac{n}{2} \\ \qquad = \left(\frac{n}{2}\right)^{n+4}} \\
& \geq 2^{n-4} \times \left(\frac{n}{2}\right)^{n+4} \\
& = \frac{n^{n+4}}{2^8}
\end{align*} Now continue by yourself.

Also this one: \(\displaystyle \displaystyle l=\lim _{n\rightarrow \infty} \frac{1+\sqrt[2^2]{2!}+\sqrt[3^2]{3!}+...+\sqrt[n^2]{n!}}{n}\).
Solution: \(\displaystyle \sqrt[(n+1)^{2}]{(n+1)!}\rightarrow \frac{1}{n} \sum _{z=1}^{n} z!^{\frac{1}{z^2 }} \) gives \(\displaystyle \lim_{n\rightarrow \infty} \sqrt[(n+1)^{2}]{(n+1)!}=\lim_{n\rightarrow \infty} \frac{1}{n} \sum _{z=1}^{n} z!^{\frac{1}{z^2 }}\)
Using AM-GM : \(\displaystyle 1\leq \sqrt[(n+1)^{2}]{(n+1)!}<\sqrt[n+1]{\frac{n+2}{2}} \)
\(\displaystyle 1\leq \lim_{n\rightarrow \infty} \sqrt[(n+1)^{2}]{(n+1)!}<\lim_{n\rightarrow \infty} \sqrt[n+1]{\frac{n+2}{2}}=1 \) so \(\displaystyle \; 1\leq L<1 \;\) gives \(\displaystyle L=1\)
\(\displaystyle \lim_{n\rightarrow \infty} \frac{1}{n} \sum _{z=1}^{n} z!^{\frac{1}{z^2 }}=1\).

Check this pdf https://www.math.ucdavis.edu/~marx/Sec. 8.6.pdf

Evaluate \(\displaystyle l=\lim_{x\rightarrow 0 } x^{x^{x}} \;\) without L'hôpital.

From now you can go deeper.
 
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Aug 2019
88
23
India
Sir, I want to ask something: In your given pdf of UC Davis there is an example which says

Find the limit
\(\displaystyle \lim_{x \to 0} \frac{\sin~4x}{x} \)

In that pdf it has been solved by graphical method, but this can be solved (as per my knowledge) like this :

\(\displaystyle \lim_{x\to 0} \frac{(\sin~4x)/4x}{x/4x} \)
As we know the numerator will be 1 as \(\displaystyle x \to 0 \)
\(\displaystyle \frac{1}{1/4}~=~ 4 \)

So, which is better? I mean, should I rely more on graphical methods or analytical methods?
 
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