So I do know how to find the answer n/m, using l'hopital's rule, but how to find it without using that rule?  $\lim _{x\to 1}\left(\frac{x^n-1}{x^m-1}\right)=\frac{n}{m}$
121778
$$\lim_{x\rightarrow 1} \frac{x^m-1}{x^n-1}$$   I found this in a book (recommended by the professor) without solutions or hints.  My problem is that I can't find definitions about this kind of problems and I have no idea how to start, any kind of help would be great. So far I've been dealing with "$\lim_{x\rightarrow ∞}$" only. Also without L'Hospital, because we haven't had this in lecture.
91502
Find the limit $$\lim_{x \to \infty}\left((x+3)^{1+1/x}-x^{1+1/(x+3)}\right).$$   I did nothing by now.
86516
Find the following limit: $$\displaystyle\lim_{x\to 0+} \left(\frac{\sin x}x\right)^{1/{x^2}}$$   Well I tried to do the $\exp\left(\frac{ \ln\frac{\sin x}{x}}{x^2}\right)$ then apply LHR but I seem to get to endless dervivations...   There's got to be a more simple approach.
12735
$$\lim_{x\to 0} \frac{\ln(1+x)}x$$  The process I  want to take to solving this is by using the definition of the limit, but I am getting confused. ( without l'hopitals rule)   $$\lim_{h \to 0} \frac{f(x+h) - f(x)}h$$  $$\lim_{h \to 0} \frac{\frac{\ln (1+x+h)}{x+h} - \frac{\ln(1+x)}x}h$$  $$\lim_{h \to 0} \frac{x\ln(1+x+h) - (x+h)\ln (1+x)}{hx(x+h))}$$  At this point I get confused because I know the answer is $1$, but I am not getting this answer through simplification of my formula.
40636
How to solve $\lim\limits_{x\to 0} \frac{x - \sin(x)}{x^2}$ Without L'Hospital's Rule? you can use trigonometric identities and inequalities, but you can't use series or more advanced stuff.
117850
When calculating the limit $L=\big( 1 - \frac 1 {\sqrt n}\big)^n$, $n\to\infty$, what allows me to do the following:  $$ L=\lim \left(\left(1-\frac {1}{\sqrt n}\right)^\sqrt{n} \right)^\sqrt{n} $$  As the term inside the outer parenthesis goes to $e^{-1}$, we have $L=\lim e^{-\sqrt n}=0$.  It's like we're distributing the parenthesis somehow:  $$\lim \left(\left(1-\frac {1}{\sqrt n}\right)^\sqrt{n} \right)^\sqrt{n}=\lim \left(\lim\left (1-\frac {1}{\sqrt n}\right)^\sqrt{n} \right)^\sqrt{n}=\lim e^{-\sqrt n}$$  The question is: Why can we do this? Which property are we using?
5779
I want to calculate the limit of: $$\lim_{x \to 0} \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x} $$ or prove that it does not exist. Now I know the result is $4$, but I am having trouble getting to it. Any ideas would be greatly appreciated.
31513
You can't use L'Hospital's rule :S  $$\lim_{x\to2} {\sqrt{6-x}-2\over\sqrt{3-x}-1}$$  I've tried to multiply by conjugates but ended up with a so complex equation, please help, anyone? :S
55505
Consider the Limit  $\frac{x+\sin x}{x-\sin x}$  as $x  \rightarrow \infty$. Clearly this limit exists and is equal to $1$. Also notice that the function is of the form  $\frac{\infty}{\infty}$ and thus, L'Hopitals rule can be applied here. But on applying L'Hopitals rule, I am getting  Limit  $\frac{1+\cos x}{1-\cos x}$ as $x  \rightarrow \infty$, which is obviously non-converging. Where did I go wrong?
123790
I have a little problem with limit of this function:  $\lim_{x \to \infty} x^2(2017^{\frac{1}{x}} - 2017^{\frac{1}{x+1}})$  I have tried de L'Hopital rule twice, but it doesn't work. Now I have no idea how to do it.
109894
How to solve this limit without using L'Hospital rule?  $$\lim_{x \rightarrow 0}{\frac{(1+x)^a-1}{x}}$$
70007
now I am evaluating limits of functions, but i dont know how to start to solve this limit. It is possible without L Hopital's rule?  $\lim_{x\to0}{{\frac{\ln(\cos(4x))}{\ln(\cos(3x))}}}$?
66681
The limit is $$\lim_{x\to0}\left(\frac{1}{\sin^2x}-\frac{1}{x^2}\right)$$ which I'm aware can be rearranged to obtain the indeterminate $\dfrac{0}{0}$, but in an attempt to avoid L'Hopital's rule (just for fun) I tried using the fact that $\sin x\approx x$ near $x=0$. However, the actual limit is $\dfrac{1}{3}$, not $0$.  In this , the approximation reasoning works out.
74526
How to find the limit of: $$ \lim_{x\to \infty }x\left(\sqrt[x]{a}-1\right)$$ Without using L'hôspital rule.  I've tried to bound the term and use the squeeze theorem but I couldn't find the right upper bound. I've also tried to convert $a^\frac{1}{x}$ to $e^{\frac{1}{x}\ln{a}}$ but it didn't help me.  Whats the right way to evaluate that limit?
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 $$f(x) = \exp \left( \ln f(x) \right)$$ ?
68437
Small question, I'm trying to solve this limit but I just can't wrap my head around this problem.  $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$  L'Hopital just seems to make it messier.   It's probably pretty simple - I'd like to hear what I'm missing.
116726
Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$