Articles of limits

Limit of $\dfrac{x}{\log x}$ as $x\to\infty$

What would be the value of $\displaystyle\lim_{x\to\infty}\dfrac{x}{\log x}$? Is it infinity or some constant value?

Prove that if $ |a_n-a_{n-1}| < \frac{1}{2^{n+1} }$ and $a_0=\frac12$, then $\{a_n\}$ converges to $0<a<1$

I try to solve this question but I don’t know how. given $ a_0 = \frac12 $ and for each $n\geq 1$: $$ |a_n-a_{n-1}| < \frac{1}{2^{n+1}} $$ show that $\{a_n\}$ converges and the limit is $a$ such that $0<a<1$ Update (Edited): I showed by cauchy that $ |a_m-a_n| < |a_m-a_{m-1}+a_{m+1}-…+a_{n+1}-a_n| < \frac{1}{2^{m+1}} + \frac{1}{2^{m}}+…+\frac{1}{2^{n+2}}$ by […]

Limit involving a hypergeometric function

I am new to hypergeometric function and am interested in evaluating the following limit: $$L(m,n,r)=\lim_{x\rightarrow 0^+} x^m\times {}_2F_1\left(-m,-n,-(m+n);1-\frac{r}{x}\right)$$ where $n$ and $m$ are non-negative integers, and $r$ is a positive real constant. However, I don’t know where to start. I did have Wolfram Mathematica symbolically evaluate this limit for various values of $m$, and the […]

Calculate limit involving binomial coefficient

How can I calculate this limit. Let $p\in[0,1]$ and $k\in\mathbb{N}$. $$\lim\limits_{n\to\infty}\binom{n}{k}p^k(1-p)^{n-k}.$$ Any idea how to do it ?

Using the definition of limit prove the following

Using the definition of limit prove that $$\lim_{x\to \infty}\frac{2x^2+x+1}{x^2-3x+1}=2$$ $$\lim_{x\to \infty}\frac{x-[x]}{x}=0$$ $[x]$ greatest integer function Please help me to solve such type of problem using $\epsilon$-$K$ method.

Limit as $x\to 0$ with zero division

I’m having trouble solving this limit: $$ \lim_{x\to 0} {x\over e^x-e^{-x}} $$ Thanks for any help. I’ve tried expanding it by $(x+1)/(x+1)$, but it didn’t help

Limit to Infinity question?

$$\lim_{x\to\infty}\left(-\sqrt{-2x+x^2}+\sqrt{2x+x^2}\right)=2$$ I’m not sure how to go about solving this problem.

Proof of convergence of $a_{n+1} = \dfrac{a_n^2 + 1}{3}$ in $\mathbb{R}$ and finding its limit

I’m starting a class on Advanced Mathematics I next semester and I found a sheet of the class’es 2012 final exams, so I’m slowly trying to solve the exercises in it or find the general layout. I will be posting a lot of questions with the exercises I find challenging, and I would like to […]

Finding functions for the squeeze theorem for $\lim_{x \to 0}{\frac {x}{\sin x}}$

I was solving a problem today and at one point I had to evaluate the $\lim_{x \to 0}{\frac {x}{\sin x}}$. I know I could easily do this with L’hôspital’s but I haven’t learned that yet. So what I did was try to use the squeeze theorem with the two bounding functions of $ f (x)=x^2 […]

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C$?

What is $\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$ for $z \in \mathbb C^*$? I need it to determine the type of the singularity at $z = 0$.