Articles of limits

Closed-form formula to evaluate $\sum_{k = 0}^{m} \binom{2m-k}{m}\cdot 2^k$

Inspired by this question I’m trying to prove that $$\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!} \approx e^{\frac{x}{2}}$$ So I needed to find the value of $$\frac{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}}{e^{\frac{x}{2}}} = \frac{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{m! (2m-k)!}{(m-k)!(2m)!}\frac{x^k}{k!}}{\lim_{m \to \infty} \sum_{k = 0}^{m} \frac{\frac{x}{2}^k}{k!}} \\ = \lim_{m \to \infty} […]

I need to understand why the limit of $x\cdot \sin (1/x)$ as $x$ tends to infinity is 1

here’s the question, how can I solve this: $$\lim_{x \rightarrow \infty} x\sin (1/x) $$ Now, from textbooks I know it is possible to use the following substitution $x=1/t$, then, the ecuation is reformed in the following way $$\frac{\sin t}{t}$$ then, and this is what I really can´t understand, textbook suggest find the limit as $t\to0^+$ […]

Why is this limit said to equal some value rather than approach that value?

I have rewritten this entire question, since what I’ve learned since asking it requires me to restate it. I want to get rid of the obfuscating revisions. Let’s say that f is a continuous function. $f(x)$ approaches L as x approaches a. So $\lim\limits_{x \to a}f(x) = L$ When it’s said that the gradient of […]

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does it matter if I further impose that $L_x$ and $L_y$ are nonrandom? I tried to replicate the argument for the nonstochastic case (included below for completeness) but I […]

Find the value of $\lim_{n\to \infty}\sum_{k=0}^n\frac{x^{2^k}}{1-x^{2^{k+1}}}$.

If $0 \lt x \lt 1$ and $$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+…..+\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then Find $\lim\limits_{n\to \infty}A_n$.

How to calculate $\lim\limits_{{\rho}\rightarrow 0^+}\frac{\log{(1-(a^{-\rho}+b^{-\rho}-(ab)^{-\rho}))}}{\log{\rho}} $ with $a>1$ and $b>1$?

How to calculate this question? $$\lim\limits_{{\rho}\rightarrow 0^+}\frac{\log{(1-(a^{-\rho}+b^{-\rho}-(ab)^{-\rho}))}}{\log{\rho}} ,$$ where $a>1$ and $b>1$. Thank you everyone.

Finding $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without L'Hopital and series expansion.

So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L’Hopital’s rule. I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.

Can $f(x,y) = |x|^y$ be be made continuous?

Can $f(x,y) = |x|^y$ be appropriately defined at (0,0) in order to be continous there . if we approach from path y=mx then f(x) becomes |x|^mx with x approaches to 0 .then by taking logs and we get infinity by infinity form . on solving with regular l’hop method ans comes cout to be e^-mx […]

taking the limit of $f(x)$, questions

How do I take the limit of the following functions? I had included some of my thoughts with them. $\lim_{x\to\infty}\dfrac{4x^3 – 2x + 1}{8x^3 + \sin(x^2) – x^{-1}}$; my thoughts: I am not sure about the bottom since there are the sine function and $-1$ power $\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}$; my thoughts: isn’t $e^x$ faster since $x^{x-1}$ is […]

How to find $\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$?

$$\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$$ How can I find this limit?