Articles of limits

Help with a limit of an integral: $\lim_{h\to \infty}h\int_{0}^\infty{{ {e}^{-hx}f(x)} dx}=f(0)$

I’m not sure how to handle limits and integral and I would like some help with the following one: let $f:[0,\infty)\rightarrow \Bbb{R}$ be a continuous and bounded function, show that $$\lim_{h\to \infty}h\int_{0}^\infty{{ {e}^{-hx}f(x)} dx}=f(0)$$ I tried many things from The fundamental theorem of calculus and define $F$ such that $F’=f$ and use integration by parts […]

Find the value of : $\lim_{x \to \infty}( \sqrt{4x^2+5x} – \sqrt{4x^2+x})$

$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} – \sqrt{4x^2+x}\ \right)$$ I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\displaystyle \frac{4x}{\sqrt{4x^2+5x} + \sqrt{4x^2+x}}$, but I can conclude nothing […]

Help with epsilon-delta proof that 1/(x^2) is continuous at a point.

I’m trying to prove that $\lim_{x \to x_0} \frac{1}{ x^2 } = \frac{1}{ {x_0}^2 }$. I know this means that for all $\epsilon > 0$, I must show that there exists a $\delta > 0$ such that $\left | x – x_0 \right | < \delta \Rightarrow \left | \frac{1}{ x^2 } – \frac{1}{ {x_0}^2 […]

A limit about $a_1=1,a_{n+1}=a_n+$

Let the sequence $\{a_n\}$ satisfy $$a_1=1,a_{n+1}=a_n+[\sqrt{a_n}]\quad(n\geq1),$$ where $[x]$ is the integer part of $x$. Find the limit $$\lim\limits_{n\to\infty}\frac{a_n}{n^2}$$. Add: By the Stolz formula, we have \begin{align*} &\mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{n^2}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{a_{n + 1}} – {a_n}}}{{2n + 1}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\left[ […]

Limit of product with unbounded sequence $\lim_{n\to\infty} \sqrt{n}(\sqrt{n}-1)=0$

This question already has an answer here: How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} – 1) = 0$? 5 answers

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$

This question already has an answer here: Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$ 4 answers

Find the limit of $(\sqrt2-\root3\of2 )(\sqrt2 -\root4\of2 )…(\sqrt 2-\root{n}\of2 )$ without using the squeezing principle

$$\mathop {\lim }\limits_{n \to \infty } (\sqrt 2 – \root 3 \of 2 )(\sqrt 2 – \root 4 \of 2 )…(\sqrt 2 – \root n \of 2 )$$ I was able easily to find the limit of this series using the squeezing principle, but how can you find it without it? I’m guessing it’s an […]

Proving that $\left(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}\right)$ has a limit

$$x_n=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}$$ How can we prove that the sequence $(x_n)$ has a limit? I have to use the fact that an increasing sequence has a limit iff it is bounded from above. No more “advanced” tools can be used. It’s obvious that this sequence is increasing, but I am having trouble finding a bound.

Problem 7 IMC 2015 – Integral and Limit

I’m trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ And here is my approach: Since $A^\frac{1}{x}=\exp\left(\frac{\ln A}{x}\right)$, then using Taylor series for exponential function, we have $$\exp\left(\frac{\ln A}{x}\right)=1+\frac{\ln A}{x}+\frac{\ln^2 A}{x^2}+\frac{\ln^3 A}{x^3}+\cdots=1+\frac{\ln A}{x}+\sum_{k=2}^\infty\frac{\ln^k A}{x^k}$$ Hence, integrating term by term is trivial. […]

Continuity of $\frac{2xy}{x^2+y^2}$ at $(0,0)$

Given a Heaviside function $$f(x,y)=\begin{cases}\frac{2xy}{x^2+y^2}, &x^2+y^2 \neq 0\\0 ,&x^2+y^2=0 \end{cases}$$ Letting $a$ and $b$ be fixed constants, show that for all values of $a$ and $b$, including $0$, the one variable functions $g(x)=f(x,b)$ and $h(y)=f(a,y)$ are both continuous on the entire real line. And how to determine whether the function is continuous at $(0,0)$. What […]