# Tagged Questions

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### Discontinuity in multivariable calculus

Show that the function F(x, y) = x(1 - cos(x - y))/(x y)^2 has removable discontinuity along the line x = y. Which values should we assign to this function on the diagonal x = y in order to turn it ...
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### Differentiability of scalar function

Let $f:\mathbb R→\mathbb R$ be a continuous function, with $f(0)=0$. Let $F(x,y)=xf(y)+yf(x)$. Analize if $F$ is differentiable at the origin. I've proved that ...
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### Solve $\lim_{(x, y) \to (4, 0)} x^2y \ln(x y)$

I got this question: $$\lim_{(x, y) \to (4, 0)} x^2y \ln(x y)$$ I know this limit does not exist by using Wolfram but I get 0 using all method. How to prove this limit does not exist?
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### Evaluating the limit of a multi-variable function

$$\lim_{(x, y) \to(0, 0)}\frac{(\sin y)(2^x - 1)}{xy}$$ I have no idea how to even approach this. I can't find a way to simplify this. Any hints would be appreciated.
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### Multivariable Delta Epsilon Proof — looking for a hint

I have the limit $$\lim_{(x,y)\to(0,0)}\frac{x^3y^2}{x^4+y^4},$$ and would like to show with an $\epsilon-\delta$ proof that it is zero. I know with a situation like ...
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### Find where the limit does not exist for the function

Given the function: $f(x,y) = \frac{xy^4}{x^2+y^8}$, find a path where the limit does not exist at the origin. I am having problems with this because of lot of paths go to $0$ but I know the limit ...
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### Is this function continuous at the origin

I have just started learning to use the two-path test to find limits and I am very doubtful of my ability so I am verifying what I have done below is correct. I first tried letting y=0 and x -> 0 ...
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### Showing limit does not exist using two-path test

I am new to using two-path test and my textbook only discusses it without showing any examples. I attempted to do this question below but I am not sure if I am correct. The question says to show the ...
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### Making a multivariable Function continuous

This function $$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$ can be made continous for $f(0,y)$ by defining $$f(0, y) = 2y .$$ My question is: how can i get to this conclusion ("$2y$ must be it") ...
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### Limits along what curves suffice to guarantee the existence of a limit?

For functions f from $R^2$ to $R$, we can define the limit of $f ( x,y)$ as $(x,y)$ goes to $(a,b)$ along the curve $C$ for any continuous curve $C$ passing through (a,b). And it is a theorem that if ...
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### How to compute $\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$?

How would one compute $$\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$$ if $a,b,c>0$. I've never done an integral 3 variables before! This is from a chapter called Interchange of ...
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### Advanced Calc 2 Interchange of Limit Operations Question

Let $a_1, a_2, a_3,...$ be a sequence of non-negative real numbers, let $S_1, S_2, S_3,...$ be a sequence (finite or infinite) of disjoint nonempty sets of natural numbers whose union is ...
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### Find the limit if it exists. Multivariable calculus

$$\lim_{(x,y)\to (4,0)}y^2\cos\left({1\over(x-4)^2+y^2}\right)$$ I am having trouble evaluating this limit. If somebody could help me, that would be awesome! Thanks!
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### I should show that $\lim \limits_{(x,y)\to(0,0)}2x\log(x^2+y^2)=0$

I should show that $$\lim_{(x,y)\to(0,0)}2x\log(x^2+y^2)=0$$ which inequality should I use?
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### is the limit continuous or not?

is $(x^2 + y^2) / (x^2 - y^2)$ continuous or not at $(0,0)$? I think it is not continuous at $(0,0)$. To check I just plugged in the points and got $0$. Did I do that right? and also is there another ...
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### Which of these statements about multivariable limits are true?

Let $f :\mathbb{R}^2 \to \mathbb{R}$ be a map. Then a) $\lim\limits_{x\to 0} \lim\limits_{y\to 0} f (x, y)$ exists implies $\lim\limits_{(x,y)\to 0} f (x, y)$ exists. b) ...
Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...