This is a math question I'm supposed to do, I know the limit does not exist(it's mentioned in the question) but I can't find any 2 paths that gives me any other answer than 0. Please if you can answer this, help me out Thanks
122683
I'm trying to find $2$ paths through $(0,0)$ to show that $\lim_{(x,y) \to (0,0)}\dfrac{xy^2}{x^2 + y^4}$ does not exist.  I only manage to use path $y=x$ but can't find second one. Any thoughts on which path to choose to show that this limit doesn't exist? Thanks!
60872
I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer, as it doesn't work for $N=1$ or $N=2$. Appreciate your help :)
62587
This is a problem occurs to me when I was trying to find the limit of a function with two variables(if only it exists). Please help.  When I have to show that the limit does not exist for some function. Then by showing along two paths have two different limits I can prove it since the functions with two variables have infinite number of directions of approach to the given point.   So selecting the paths, should I only take the linear ones or can I take non linear paths also?  As an example,consider the limit of $\frac{xy^3}{x^2+y^6}$ as $(x,y)$ goes to $(0,0)$ Then taking along $x$ axis I have the limit $0$. Can I take $y=x^{1/3}$ which gives me the limit $1/2$ ?  When I check the answer on some website it seems they haven't consider curved functions to find different limits.  Please explain. Where am I wrong?
24947
When $\lim_{n\to\infty} x_n = a$, and $\lim_{n\to\infty} y_n = b$, find the limit, $$\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}.$$ Thank you for your help in advance.
77486
How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks!
16307
This is a true/false question.     If the limits $$\lim_ {x\to a}\ f(x)\quad \text{ and }\quad  \lim_ {x\to a}\ g(x)$$ don't exist, limits $$\lim_ {x\to a}\ (f(x)+g(x))\quad \text{ and }\quad \lim_ {x\to a}\ (f(x)g(x))$$ don't exist also.   My idea is, that, because $$\lim_ {x\to a}\ (f(x)+g(x))=\lim_ {x\to a}\ f(x)+\lim_ {x\to a} g(x)$$ then the limits doesn't exist, correct?
20697
I am trying to find the limit of:  $$\lim_{n \to \infty} n(\sqrt[n]{2} - 1)$$  I know it should be very simple but I don't seem to get it. Thanks in advance for any help!
14445
$ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $  The answer should be $ {e}^\frac{1}{3} $  I haven't encountered problems like this before and I'm pretty confused, thank you.  I guess we must use the remarkable limit of $ \frac{\tan(x)}{x} $ when $ x $ approaches $ 0 $ by dividing and then multiplaying with $x$.
41522
(This was asked due to the comments and downvotes on this  answer. I am not that good at maths, so was wondering if I had made any basic mistakes)  Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is undefined:     $x = \frac00$    $x \cdot 0 = 0$        Hence There are an infinite number of values for $x$ as anything multiplied by $0$ is $0$.   However, it seems to have got comments, with two general themes.  Once is that you lose the values of $x$ by multiplying by $0$.  The other is that the last line is:      $x \cdot 0 = \frac00 \cdot 0$   as it involves a division by $0$.  Is there any merit to either argument? More to the point, are there any major flaws in my explanation and is there a better way of showing why $\frac00$ is undefined?
70412
I read in Stewart "single variable calculus" page 83 that the limit $$\lim_{x\to 0}{1/x^2}$$ does not exist. How precise is this statement knowing that this limit is $\infty$?. I thought saying the limit does not exist is not true where limits are $\infty$. But it is said when a function does not have a limit at all like  $$\lim_{x\to \infty}{\cos x}$$.
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.
122021
Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists ? If it does then what is the value ?   I don't think even L'Hospital's rule can be applied . Please help . Thanks in advance
117031
$$\lim\limits_{x\to\infty} \ \frac{\ln \ x}{x}  = 0$$  How can I know this without looking at the graph? What's the easiest way to equate that expression to 0?  I understand that an integer over x approaches 0 as x approaches 0, but since ln x is an increasing function as x approaches infinity, how do we know which effect outweighs the other?
18749
Study the following limit: $$ \lim\limits_{ (x,y,z) \to (0,0,0) } \frac{xy-z^2}{x^2+y^2+z^2} $$   I'm not sure if this way of proving that the limit does not exist is correct so I'm this is basically for a proof verification.  Approaching $(0,0,0)$ through the plane $z=0$, then the limit becomes  $$ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $$  Now this limit does not exist, because approaching $(0,0)$ through the line $y=0$, the limit becomes  $$ \lim\limits_{ x \to 0 } \frac{0}{x^2} =0$$  But if we approach $(0,0)$ through the line $y=x$ we have that  $$ \lim\limits_{ (x,x) \to (0,0) } \frac{x^2}{2x^2}=\lim\limits_{ (x,x) \to (0,0) } \frac{1}{2} =\frac{1}{2}$$  So the limit $ \lim\limits_{ (x,y) \to (0,0) } \frac{xy}{x^2+y^2} $ doesn't exist. This means that the limit we began with doesn't exist, correct?
112239
$$ \begin{cases}xy^2\sin\left(\frac 1 y\right), \text {if $y$ $\neq$ 0} \\ 0, \text{if $y$ = 0} \end{cases}$$  I'm not sure how to work out this limit as $x$ goes to infinity. If it were just the y part, I know that $y^2$ goes to 0 and $\sin$ is a limited function, so it has to go to 0, but I don't know what to do with the $x$ term.
15740
Ratio and root tests won't help. And I can't use the comparison test because $ |a_n| $ is not necessarily smaller than $ n^2a_n $.   Can I use limits? We know:    $\lim\limits_{n \to \infty} a_n = 0 $ $\lim\limits_{n \to \infty} |a_n| \ne 0 $   And we need to prove:    $\lim\limits_{n \to \infty} n^2a_n \ne 0 $   Any ideas/hints?
36258
Let $0 &lt; x_1 &lt; 1$ and $x_{n + 1} = x_n - x_n^{n + 1}$ for $n \geqslant 1$.      Prove that the limit exists and find the limit in terms of $x_1$.   I have proved the existence but cannot manage the other part.  Thanks for any help.