Articles of limits

How can I evaluate $\lim_{x\to1}\frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}$ without invoking l'Hôpital's rule?

In the math clinic I work at, somebody in a Calculus 1 class asked for help with this limit problem. They have not covered basic differentiation techniques yet, let alone l’Hôpital’s rule. $$\lim_{x\to1}\frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}$$ We have tried various algebraic techniques, such as multiplying the top and bottom of the fraction by the conjugate of the denominator, […]

Find the value of : $\lim\limits_{n\to \infty} \sqrt {\frac{(3n)!}{n!(2n+1)!}} $

First of all, sorry if something similar to this has been posted before (it’s my first time in this web). I need to calculate the limit as $n\rightarrow \infty$ for this: $$\lim\limits_{n\to \infty} \sqrt [n]{\dfrac{(3n)!}{n!(2n+1)!}} $$ But I don’t know which steps I need to follow in order to do it. Thank you everyone in […]

Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?

How find the limit $\lim_{n\to+\infty}\sum_{i=2}^{n}\dfrac{\ln{i^2}}{i^2}$

find the $$\lim_{n\to+\infty}\left(\dfrac{\ln{2^2}}{2^2}+\dfrac{\ln{3^2}}{3^2}+\dfrac{\ln{4^2}}{4^2}+\cdots+\dfrac{\ln{n^2}}{n^2}\right)$$ My try: $$\lim_{n\to+\infty}\left(\dfrac{\ln{2^2}}{2^2}+\dfrac{\ln{3^2}}{3^2}+\dfrac{\ln{4^2}}{4^2}+\cdots+\dfrac{\ln{n^2}}{n^2}\right)=2\sum_{n=2}^{\infty}\dfrac{\ln{n}}{n^2}$$ and I know solve this following $$\sum_{n=2}^{\infty}(-1)^n\dfrac{\ln{n}}{n}=\ln{2}\left(C-\dfrac{\ln{2}}{2}\right)$$ where $C$ is Euler constant Solution:note this following $$\lim_{n\to\infty}\left(\dfrac{\ln{1}}{1}+\dfrac{\ln{2}}{2}+\cdots+\dfrac{\ln{n}}{n}-\dfrac{(\ln{n})^2}{2}\right)=l$$ we let $$S_{n}=\sum_{k=1}^{n}\dfrac{(-1)^k\ln{k}}{k}$$

How can I calculate the limit of exponential divided by factorial?

This question already has an answer here: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$. 13 answers

Help solving the limit $\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n}$

$$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} $$ I’ve tried: $$\lim_{n\to \infty} \left(1+\frac{1}{n!}\right)^{2n} = \lim_{n\to \infty} e^{\ln{\left(1+\frac{1}{n!}\right)^{2n}}} = \lim_{n\to \infty} e^{2n\ \ln{\left(1+\frac{1}{n!}\right)}}$$ But I don’t know how to work with the factorial

Uniformly bounded sequence of holomorphic functions converges uniformly

Consider an open connected set $\Omega\subset \mathbb{C}$, and $f_n\subset H(\Omega)$. Suppose $f(z)=\lim_{n\to\infty}f_n(z)$ exists and $|f_n(z)|\leq M$ for all $z\in \Omega$. Show that $$\lim_{n\to\infty}\sup_{z\in K}|f_n(z)-f(z)|=0$$ for any compact $K\subset \Omega$. Note: If correct, I am sure this is a well known result; but I don’t see why it’s correct, and I haven’t found a reference, either.

limit of $\sqrt{\frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots (2n)}}$ using geometric mean

How to find this question using Geometric Mean? $$\sqrt[n]{\frac{1\cdot 3\cdots (2n-1)}{2\cdot 4\cdots (2n)}}$$ Thanks!

$\epsilon$-$\delta$ proof that $f(x) = x \sin(1/x)$, $x \ne 0$, is continuous

I’m doing an exercise that asks me to prove that $f$ is continuous using a $\epsilon$-$\delta$ proof. I have that $$ f(x) = \begin{cases} x\cdot \sin \frac1x,&x\neq 0 \\ 0,&x = 0 \end{cases} $$ I’ve already managed to show this property for $x=0$. How can I show it for $x \ne 0$, also using a […]

Incorrect method to find a tilted asymptote

Suppose I want to find the slanted asymptote for the graph of $\displaystyle y=\frac{x^2+x-6}{x+2}$. Using division, we have $\displaystyle y=x-1-\frac{4}{x+2};\;$ so $y=x-1$ is the slanted asymptote. I would like to find out, though, what is wrong with the following incorrect way of finding the asymptote: $\displaystyle y=\frac{x^2+x-6}{x+2}=\frac{x+1-\frac{6}{x}}{1+\frac{2}{x}}\approx\frac{x+1}{1}=x+1$, so $y=x+1$ is the slanted asymptote.