Articles of limits

Limit of a summation, using integrals method

I have seen an interesting question on stackexchange, which I would like to requote so that I can understand the answer =) $\lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ How can the numerator be expressed as an integral? $= \lim\limits_{n\to\infty} \dfrac{1^{99} + 2^{99} + \cdots + n^{99}}{n^{100}}$ $=\lim\limits_{n\to\infty} \frac{\sum\limits_{k=1\to n} k^{99} }{n^{100}}$ $=\lim\limits_{n\to\infty} \frac{\sum\limits_{k=1\to […]

Please prove: $ \lim_{n\to \infty}\sqrt{\frac{1}{n!}} = 0 $

Possible Duplicate: $ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite Please prove: $$ \lim_{n\to \infty}\sqrt[n]{\frac{1}{n!}} = 0 $$

Difficulty in evaluating a limit: $\lim_{x \to \infty} \frac{e^x}{\left(1+\frac{1}{x}\right)^{x^2}}$

The limit I have to evaluate is this – $$\lim_{x \to \infty} \frac{e^x}{\left(1+\frac{1}{x}\right)^{x^2}}$$ I first checked if L’Hopital’s rule applies here. The limit of both numerator and denominator is $\infty$. But differentiating the denominator yield a even more complicated expression. I am not getting how to approach this question using some other method. Thank you.

limit of the sum $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n} $

This question already has an answer here: The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$ 10 answers

Limit the difference between the sinuses.

What is wrong? $$ \lim_{n\rightarrow\infty}(\sin(\sqrt[3]{n^3-9})-\sin(n)) = $$ $$ = \lim_{n\rightarrow\infty}2\sin(\frac{\sqrt[3]{n^3-9}-n}{2})\cos(\frac{\sqrt[3]{n^3-9}+n}{2}) = $$ $$ = 2\lim_{n\rightarrow\infty}\sin(\frac{n(\sqrt[3]{1-\frac{9}{n^3}}-1)}{2})\cos(\frac{n(\sqrt[3]{1-\frac{9}{n^3}}+1)}{2}) = $$ $$ = 2\lim_{n\rightarrow\infty}\sin(\frac{n*0}{2})\cos(\frac{n*2}{2}) = $$ $$ = 2\lim_{n\rightarrow\infty}\sin(0)\cos(n) = 0; $$ But wolfram get another answer.

Riemann Sum. Limits.

So how to approach this one? $\frac1n\sum g(\frac{r}{n}) $ . How to convert in this form? As I can see r and n will have different powers.

Prove that $\lim \limits_{n\to\infty}\frac{n}{n^2+1} = 0$ from the definition

This is a homework question: Prove, using the definition of a limit, that $$\lim_{n\to\infty}\frac{n}{n^2+1} = 0.$$ Now this is what I have so far but I’m not sure if it is correct: Let $\epsilon$ be any number, so we need to find an $M$ such that: $$\left|\frac{n}{n^2 + 1}\right| < \epsilon \text{ whenever }x \gt […]

Conjecture about $A(z) = \lim b^{} ( c^{} (z) ) $

Let $b(z),c(z)$ be analytic on the strictly positive reals. Let $^{[*]}$ denote composition. Conjecture : If $A(z) = \lim b^{[n]} ( c^{[n]} (z) ) $ Such that 1) the limit ( $A(z)$ ) exists for all strictly positive real $z$. 2) the sequence $a_n(z) = b^{[n]} ( c^{[n] } (z) ) $ is bounded in […]

Limit of a sequence of determinants.

Let $\beta>0$ be given. For each $n\geq 2$, let $\Delta_n=\det M_n$ denote the determinant of the following matrix: \begin{align} M_n = \begin{pmatrix} 2+\epsilon^2 & -1 & 0 & 0 & 0 & -1 \\ -1 & 2+\epsilon^2 & -1 & 0 & 0 & \ddots \\ 0 & -1 & 2+\epsilon^2 & -1 & 0 […]

Real Analysis – Prove limit $\lim_{x\to 25} \sqrt x = 5$

Prove that $\lim\limits_{x\to 25} \sqrt x = 5$ using $\epsilon$ and $\delta$. can someone check my work please? $$|x-a|<\delta \qquad\Rightarrow\qquad |f(x)-L|<\varepsilon$$ To prove this we must show the following: There exists a $\delta>0$ for every $\varepsilon>0$ such that $$|x-25|<\delta \qquad\Rightarrow\qquad |\sqrt x-25|<\varepsilon.$$ \begin{gather*} -\varepsilon < \sqrt x – 25 < \varepsilon \\ 25-\varepsilon < \sqrt […]