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.

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.

Could someone please explain

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.

Thanks in advance

Central Limit Theorem for Wilcoxon signed rank tests

Let $X_i, i=1,2,…,n$ be a set of iid observations, assumed symmetric about $mu$. Let $R_i$ be the rank of the absolute deviations from some $mu_0$, i.e. $R_i=text{rank}(|X_i-mu_0|)$. Let $Z_i=text{sign}(X_i-mu_0)$. Under the null hypothesis $mu_0=mu$.

If I pick up only one sample each time, let X be the random variable, the expected average value E(X) is 0.5, isn’t it? the variance of X is 0, isn’t it? so how
how can I apply CLT for signed rank tests?