So i'm a little shaky on limits and I want to show  I could take the limit of$  \frac{x^{2.5}}{100^{x}} $  but it would get very messy with L'Hopitals rule is their an easier way to show that this goes to 0 without having to do  a bunch of algebra?
1279
In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem.
40617
$$ \lim_{(x,y)\to (0,0)} \frac {x^3y^2}{x^4+y^6}  $$ Does this limit exist?   I've tried substituting y=x^0.5 and y=x^(2/3) which both goes to 0.
70265
I have passed the calc sequence, and thought that I understood things.  But I have trouble with limits now and again.  From the way I understand it, the 'best' way to evaluate a limit, is to evaluate the expression at the number in the limit.  In that way it's not only a limit, but an algebraic expression (so, I think, the need for limits goes away entirely, superseded by accuracy.)  But then I see something like $\lim_{x\to 2} \frac{x^2-4}{x-2}$.  Now, I think in my mind 'the best way to evaluate this is just to plug in $2$'  But it doesn't work that way, as the limit is $4$, and if I plug in, the answer is $0/0$.  I think that people here, with their natural talent in math, can perceive where my train of thought is going wrong, and correct me.  Please!
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.
76057
How to integrate this?     Evaluate   $$\lim_{x\to0}\frac{\displaystyle\int_0^x(x-t)\sin t^2\ dt}{x\sin^3x}.$$   I've had difficulty in using L'Hopital rule. At the same time failed to really understand how to differentiate or evaluate the limit of the $x\sin^3x$ and $\int_0^x(x-t)\sin t^2\ dt$  Would like to appreciate your help
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).
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?  $\lim_{x\to0}{{\frac{\ln(\cos(4x))}{\ln(\cos(3x))}}}$?
66968
I need to evaluate the limit without using l'Hopital's rule.  $$\lim_{x \to 0}{\frac{\sqrt{1+x\sin{x}}-\sqrt{\cos{2x}}}{\tan^{2}{\frac{x}{2}}}}$$
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?
110601
$$\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$$  I'm not entirely sure how to go about this, since rationalizing anything doesn't quite work for me. Just looking for some help.
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.
99286
The limit is    $$\lim\limits_{x\to0}\frac1{\ln(x+1)}-\frac1x$$  The problem is I don't know if I can calculate it normally like with a change of variables or not. Keep in mind that I'm not allowed to use L'Hôpital's rule nor the $\mathcal O$-notation.
19643
The question is rather:      Prove the following:   $$  \lim_{x \to \infty} \frac{x ^ {\sqrt x} }{ 2 ^ x} = 0. $$   I was thinking of using the Squeeze Theorem (might not be the right way to go), but finding an upper-bound function proved to be quite tricky.
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.
59001
How to show if the limit $$ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$$ exists? I suspect that there is, as I can't find any path that would show that limit doesn't exist, and WolframAlpha also suggests that the limit is (0,0).  In general, can you recommend any tips how to learn to approach similar limit problems (fractions and polynomials like this)? edit That is, excluding the polar coordinate conversion method?
69608
I know with l'Hopital's rule it becomes $-\sin(x)$ which has the limit $0$. However, I have been wondering how to evaluate this limit without l'Hopital's rule.
123897
This problem comes from our text book.   Evaluate  $$ \lim\limits_{x \to 0}\frac{2^x-1-x}{x^2} $$ without using either L'Hopital's rule or Taylor series.  The picture below shows the solution given by the text book.    Many of the people who responded to this question say that the limit diverges. But according to the solution (see the attached image) given by the text book the limit is  $ \frac{(\ln2)^2}{2} $. Is there an error in the solution given in the attached image?  Note: This problem ( and the solution) comes from Cengage, a long time favourite of the students preparing for IIT JEE  Exam (IITs are India's most prestigious engineering institutes.)  It is suprising that no student/teacher has found this mistake all these days. However, once posted on Stackexchange, the fallacy was resolved in a very short time.  Thanks to all the people who responded to this problem. Now the mistake in the book's solution is found.