## Limit of Convergent Sequences in a Compact Graph when the Domain is Compact

Suppose $$f: E to N$$ where $$E$$ is compact. The graph of $$f$$ is:
$$G(f)={(x,f(x)):xin E}$$
Also, assume that $$G$$ is compact.

Consider $$(x_n) subseteq E, (x_n)to x$$ and the corresponding sequence $$(x_n,f(x_n)) subseteq G$$. Since $$G$$ is compact, pick a subsequence:
$$(x_{n_k},f(x_{n_k})) to (x,a)$$

Question: Is $$a=f(x)$$?

## Limit distribution equal to Dirac delta

This is the problem 6.19 from the book Distributions and Operators, Gerd Grubb. I already have done parts (a) and (b).

The part (a) of this problem is proving that for $$rin(0,1]$$, the sequence $${frac{1}{2pi}sum_{n=-N}^{N}r^{|n|}e^{inx}}_{Ninmathbb{N}}$$ converges to a distribution $$P_{r}$$ in $$D'((-pi,pi))$$ and that $$P_{1}=delta$$. Part (b) is just showing that $$rmapsto P_{r}$$ is continuous.

Now, for part (c), I have to show that when $$r$$ converges to $$1$$ from the left, then $$int_{-pi}^{pi}frac{1-r^2}{1-2rcostheta+r^2};varphi(theta);dtheta$$ converges to $$varphi(0)$$ for any $$varphiin C_{0}^{infty}((-pi,pi))$$.

I thought about this for a while but I don’t have a clue. Thanks for the help.

## Show that the limit does not exist \$lim_{(x, y) to (0,0)}frac{5x^2}{x^2 + y^2}\$

Show that the limit does not exist $$lim_{(x, y) to (0,0)}frac{5x^2}{x^2 + y^2}$$

attempt:

let $$y = 0$$

$$lim_{x to 0} frac{5x^2}{x^2 + 0^2} = 5$$

let $$x = 0$$

$$lim_{y to 0} frac{5(0)^2}{y^2} = 0$$

$$5 neq 0$$, therefore two different values, limit does not exist

right?

## For each of the following evaluate the limit or show that the limit does not exist \$lim_{(x, y) to (0,0)}…

For each of the following evaluate the limit or show that the limit does not exist

$$lim_{(x, y) to (0,0)} frac{sin(x-y)}{||(x, y)||}$$

Solution:

$$=lim_{(x, y) to (0,0)} frac{sin(x-y)}{sqrt{x^2 + y^2}}$$

If we let $$y = x$$, then

$$=lim_{x to 0)} frac{sin(0)}{sqrt{2x^2}} = 0$$ [Wouldn’t this be indeterminate $$0/0$$?]

If we let $$y = -x$$, then

$$lim_{x to 0^+} frac{sin(2x)}{sqrt{2x^2}} = lim_{x to 0^+} frac{sqrt{2}sin(2x)}{2x} = sqrt{2}$$

and since they are different values we know the limit does not exist.

## Help with the following limit

I have the following limit

$$lim_{utoinfty} frac{-bu^{a-1}e^u}{e^{be^u}}$$

where a and b are constants

I have tried L’Hopital and i keep getting undefined results. I have also tried Series expansions but nothing.

Could anyone take me through this , and how i am supposed to think when tackling a problem such as this?

Thank you very much for your help and time.

## Show that the limit does not exist \$frac{5x^2}{x^2 + y^2}\$

Show that the limit does not exist $$frac{5x^2}{x^2 + y^2}$$

attempt:

let $$y = 0$$

$$lim_{x to 0} frac{5x^2}{x^2 + 0^2} = 5$$

let $$x = 0$$

$$lim_{y to 0} frac{5(0)^2}{y^2} = 0$$

$$5 neq 0$$, therefore two different values, limit does not exist

right?

## Limit when \$z\$ goes to \$0\$ of \$re(z) / im(z)\$?

It’s clear to me that this limit does not exist, because you can go to zero using the identity line and you get $$1$$, but if you go to zero in direction getting close to the imaginary line then you get $$0$$.

How can i formalize that idea, especially going to zero from the identity line?

## Evalutation of the limit \$lim_{x rightarrow + infty} (x – x^2 * ln(1+ frac 1 x))\$

I was trying to evaluate the limit $$lim_{x rightarrow + infty} (x – x^2 * ln(1+ frac 1 x))$$ without using neither Taylor series nor De L’Hopital rule, but just with notable limits such as $$lim_{x rightarrow 0} frac {e^x – 1} x = 0$$ or substitution.

I tried for a lot of times with different substitutions and notable limits, but I couldn’t find any solution.

Can you give me some hints.