How do I prove that this limit does not exist?   $$\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $$  My attempt:  When you approach from from left towards zero , say i take -0.00000000000001 . i substitute in expression i get (-1) . But if i take 0.000000000001 and substitue i get (=1) (by applying L'Hop) . But if i dont do this and apply L'Hop straighaway i get 1 .
121707
How do I prove that this limit does not exist?   $$\lim_{x\rightarrow 0} \frac{e^{1/x} - 1}{e^{1/x} + 1} $$  My attempt:  When you approach from from left towards zero , say i take -0.00000000000001 . i substitute in expression i get (-1) . But if i take 0.000000000001 and substitue i get (=1) (by applying L'Hop) . But if i dont do this and apply L'Hop straighaway i get 1 .
38160
I was recently thinking about how I could evaluate the famous limit of 'e' as I haven't ever seen a proof. I can't really find anything online so I've tried to evaluate the limit myself. And I was also thinking it would be nonsensical to use L'Hopital's rule, am I right?  So I did the following:  $$\lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^{n} = \exp \left(\lim_{n \to \infty} n \cdot \ln \left(1 + \frac{1}{n} \right) \right)$$  $$=\lim_{n \to \infty} \exp \left( n \cdot \left( \frac{1}{n} - \frac{(1/n)^{2}}{2} + \cdots \right) \right)$$  $$= \lim_{n \to \infty} \exp \left( 1 - \frac{(1/n)}{2} + \cdots \right)$$  $$= e$$  I am not sure, is my logic correct or does it create circularity by taking logarithms and assuming $$f(x) = \exp \left( \ln f(x) \right)$$ ?
15599
Given $\frac{a}{c} + \frac{b}{d} = 1$ Prove that $$\lim_{(x,y)\rightarrow(0,0)} \frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}}$$ does not exist.  So I have done the proof for strict inequalities. And the limit only exists when the fractions add up to an integer greater than $1$. I'm not sure how to approach this other than counter examples.
57957
Purely out of curiosity, is it possible to compute the following limit without using L'Hopital?  $$\lim_{x\to 0} \frac{e^{-1/x^2}}{x}$$  Using L'Hopital and realizing that we cannot get what we want if we simply take derivatives of $e^{-1/x^2}$ and $x$ directly, we have that  $$\lim_{x\to 0}\frac{e^{-1/x^2}}{x}=\lim_{x\to 0}\frac{1}{x}\frac{1}{e^{1/x^2}} = \lim_{x\to 0} \frac{1/x}{e^{1/x^2}}=0$$  I am curious about whether there is a different way of computing this.
68075
Problem: Find with proof limit of the sequence $a_n=1+\frac{1}{a_{n-1}}$ with  $a_1=1$  or show that the limit does not exist.  My attempt:  I have failed to determine the existence. However if the limit exists then it is easy to find it.  The sequence is not  monotonic and I have failed to find any monotonic subsequence subsequence. The sequence is clearly bounded below by $1$. I have observed that the sequence is a  continued fraction so it alternatively increases and decreases.   So, please help me.
12958
It seems i'm not gonna find an answer for this question which does not use LHopital's rule. How do you prove limit of (lnx)/x approaches 0 as x moves toward infinity WITHOUT using LHopital's rule?
7735
Is it possible to determine the limit  $$\lim_{x\to0}\frac{e^x-1-x}{x^2}$$  without using l'Hopital's rule nor any series expansion?  For example, suppose you are a student that has not studied derivative yet (and so not even Taylor formula and Taylor series).
59709
How do I prove that the limit of the sequence exists: $$a_n = \frac1n +\frac 1{n+1} + \frac1{n+2} + \dots + \frac1{2n}$$  I can find the limit as n goes to infinity, $\frac1{2n}$ goes to $0$, so the limit is $0$ right? But how do I prove that the limit exists? Do I have to use the $\varepsilon$ $\delta$ definition of a limit?    For part b) I have to show that the limit is less then $1$ but not less then $\frac12$, I thought the limit was $0$?
108186
I'm struggling to find the solution of the following:  $$\lim_{x\rightarrow\infty}(\sqrt{x^2+4x} - x)$$  I come to the answer of $0$.  The book has an answer of $4/1$.  The book explains a part of the question briefly. Looking at the brief answer information given I then come to an answer of $2$...  What is the answer?
44598
Given the following limit,  $$\begin{align} \lim_{x\to 0}\frac{e^{-1/x^2}-0}{x-0}\\\\ &amp; \end{align}$$  How do I calculate it? when pluggin in 0 I would get $\frac{0}{0}$
2235
Can someone show me how to calculate the limit:  $$\lim_{x \to \infty }\left(\frac{1}{e}\left(1+\frac{1}{x}\right)^x\right)^x $$  I tried to use taylor series but failed.  Thanks
39067
Is there a way of showing this limit without using L'Hopital's rule? Specifically using the substitution $u = -\log{x},$ with knowledge of the limit of $\lim_{x\to0+}\log{x}.$ Thank you.
46576
What is the limit of $0/x$ as x goes to $0$, without using L'Hopital's rule? Clearly it should be $0$, but I'm not sure how it is any different from something like the divergence of $1/r^2$ which yields a dirac delta.
54701
I am stuck trying to understand how to prove that the limit of $\sin(\frac{1}{x})$ as $x$ approaches $0$ does not exist. a hint was given: a limit does not exist if there exists an $\varepsilon &gt; 0$ such that for every $\eta&gt;0$ there exists an $x$ with $0&lt;|x-a|&lt;\eta$ whenever $|f(x)-L|\geq \varepsilon$.   so I approached this by stating that $|x|&lt;\eta$ then $\sin(\frac{1}{x})-L\geq \varepsilon$. then I set $\varepsilon \leq|f(x)-L|&lt;\sin(\frac{\eta}{2x})-L$ then used inverse sine to try and get delta alone... but now I'm looking at it trying to figure out what i did and why i did it.   can someone please walk me through a proof that this limit does not exist and maybe try and explain it to me. I am lost.
21544
Help with Limit: $$\lim_{n \to \infty} \frac{(2 n)! e^n (n)^n}{n! (2 n)^{2 n}}$$  EDIT: Posted a similar question just yesterday but bad formating from my part dropped a term, and lead me to strange places. Honestly, I am just plain stuck, I have been hitting my head against it for $2$ days straight. I know the solution should be $2^{-1/2}$ but...  Help would be appreciated. If anyone has a Wolfram Alpha PRO account I am sure that thing churns out the solution as it is something quite basic I just don't see it and its driving me crazy.
47702
I'm trying to formalize the following limit:  $$\lim_{x \to \infty} \left[1+\frac{a}{x}+o\left(\frac{1}{x}\right) \right]^x= e^a$$ however I can't see how to rigorously show this identity.  Any suggestions?
16328
I need help solving this problem. I tried L'hospital and rearranging but nothing worked. $$ \lim_\limits{x→∞} x^2\left(\ln\left(1 + \frac{1}{x}\right)- \frac{1}{x+1}\right) $$