How would one derive $\lim_{x\rightarrow0}\left(\frac{\sin(x)}{x}\right)$ without using any derivatives?
10916
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated.
113367
I want to find the limit of the following expression: $$\lim_{x \rightarrow \infty} \left(\left(\frac{x+1}{x-1}\right)^x -e^2\right){x}^2.$$ I got the limit of the first term as $e^2$ so the problem now becomes equivalent to evaluating a $0 \times \infty$  limit, but I'm having difficulty proceeding after this. I tried applying L'Hospital's rule but the derivative gets really complicated after a while. I also tried applying series expansions but that too got me nowhere. Could anyone please tell me how to proceed with this problem?
116726
Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$
30089
Find the limit:   $$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$  I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
72949
How do I prove that the derivative of $\sin$ is $\cos$ ?  This part is easy:  $$ \begin{align*} \sin'(x) &amp;= \lim_{\Delta x \to 0} \frac{\sin(x + \Delta x) - \sin(x)}{\Delta x}  \\ \sin'(x) &amp;= \lim_{\Delta x \to 0} \frac{\cos(x)\sin(\Delta x) + \sin(x) \cos(\Delta x) - \sin(x)}{\Delta x}  \\ \sin'(x) &amp;= \lim_{\Delta x \to 0} \left(\cos(x)\frac{\sin(\Delta x)}{\Delta x} + \sin(x)\frac{\cos(\Delta x) - 1}{\Delta x}\right)  \\ \end{align*} $$  but where do I go from here?
40101
Find the limit: $$\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$$   I have no idea what to do, but I thought that this is the limit of the derivative of $f(x)=\left(1+x\right)^{1/x}$, as $x$ tends to 0. Any help?
40892
Is it always possible to find the limit of a function without using  or ?  For example,   $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$  $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$  $$\lim_{x\to0}\frac{\ln(1+x)-x}{x^2}$$  $$\lim_{x\to0}\frac{e^x-x-1}{x^2}$$  $$\lim_{x\to0}\frac{\sin^{-1}x-x}{x^3}$$  $$\lim_{x\to0}\frac{\tan^{-1}x-x}{x^3}$$
116995
$$\lim_{x\rightarrow +\infty}\sqrt{x}e^{-x}\left(\sum_{k=1}^{\infty}\frac{x^{k}}{k!\sqrt{k}}\right)$$ Any hint will be appreciated.  Note: There is a related question on MathOverflow: . The MO question references this question and it also links to some other posts containing similar expressions.
45937
$$\lim_{x\to\infty}\left(\sin{\frac 1x}+\cos{\frac 1x}\right)^x$$  It is about the $(\to1)^{(\to\infty)}$ situation. Can we find its limit using the formula $\lim_{x\to\infty}(1+\frac 1x)^x=e$? If yes, then how?
115761
I am trying to find this limit:  $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$  I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you
49258
The limit is   $$\lim_{x\rightarrow0} \frac{x-\sin_n(x)}{x^3},$$  where $\sin_n(x)$ is the $\sin(x)$ function composed with itself $n$ times:  $$\sin_n(x) = \sin(\sin(\dots \sin(x)))$$  For $n=1$ the limit is $\frac{1}{6}$, $n=2$, the limit is $\frac{1}{3}$ and so on.  Can we define a recurrent relation upon that given hypothesis? Also, how do I involve $n$ into calculation, because the final limit will depend on it?  Any suggestions on how to tackle this?  Thank you!
18710
Please Help me derive the derivative of the absolute value of x using the following limit definition. $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ I have no idea as to how to get started.Please Help.  Thank You
71280
I was given a task to find  $$\lim_{x\to0}\frac{x-\sin{x}}{x^3}$$ at my school today. I thought it was an easy problem and started differentiating denominator and numerator to calculate the limit but the teacher then said we aren't allowed to use  L'Hopital's rule, but to "play around" with known limits and limit definition. I got stuck here since I can't really think of a way to do this, and according to my teacher, there are at least 4 ways. A subtle hint would be enough. Thanks.
32691
$$\lim_{x\to0} \frac{x-\sin x}{x-\tan x}=?$$ I tried using $\lim\limits_{x\to0} \frac{\sin x}x=1$. But it doesn't work :/
90394
Compute the limit:  $$\lim_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots\left(1+\frac{n}{n}\right)^{\frac{1}{n}}$$
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.
84750
Use epsilon-delta definition of limit  To show :  $\displaystyle \lim_{x \to 0}  x\sin \left(\dfrac{1}{x}\right) = 0$   How to explain this in detail with step by step explanations?