Articles of limits

Easy way to find out limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$?

What’s an easy way to find out the limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$? I don’t think binomial expansion like with $\left (1-\frac{1}{n^2} \right )^n = \left (1+\frac{1}{n} \right )^n \cdot \left (1-\frac{1}{n} \right )^n$ is possible. And Bernoulli’s inequality only shows $\left (1+\frac{1}{n^2} \right )^n \geq 1 + […]

Asymptotic behavior of $\sum\limits_{n=1}^{\infty} \frac{nx}{(n^2+x)^2}$ when $x\to\infty$

This question already has an answer here: A tricky sum to infinity 1 answer

How to solve the limit? $ \lim_{x \to +\infty} \frac{(\int^x_0e^{x^2}dx)^2}{\int_0^xe^{2x^2}dx}$

This question already has an answer here: Compute $\lim_{x \rightarrow +\infty} \frac{[\int^x_0 e^{y^{2}} dy]^2}{\int^x_0 e^{2y^{2}}dy}$ 2 answers

If $I_n=\sqrt{\int _a^b f^n(x) dx}$,for $n\ge 1$.Find with proof $\lim _{n \to \infty}I_n$

Let $a,b \in \mathbb R ,a<b$ and let $f:[a,b] \to [0,\infty) $ a continuous and non constant. attempt,using Reimann series $I_n=\sqrt [n]{\int _a^b f^n(x) dx}$ $I_n=\sqrt [n]{\lim _{k \to \infty}\sum_{i=1}^k f^n(x_i^*) dx}$ for each $i=1,2,3…n$ $x_i\in [x_{i-1},x_i]$ so our question is what is $I_n=\lim_{n\to \infty}\sqrt [n]{\lim _{k\to \infty}\sum_{i=1}^k f^n(x_i^*) dx}$

Why can we modify expressions to use limits?

If I wanted to take the limit of $\frac { \left( x\cdot \cos { \left( x \right) +\sin { \left( x \right) } } \right) }{ x+{ x }^{ 2 } } $ as x approaches 0, I cannot do it directly as that would result in dividing by zero. However, if I modify the […]

Evaluating $\lim_{x \to e} \left ( \ln(x)\right )^{1/(x – e)}$ with substitutions

I evaluated the limit with two substitutions: $$\begin{align} L&:=\lim_{x \to e} \left ( \ln(x)\right )^{1/(x – e)}=\\ &=\begin{bmatrix} t = x – e\\ x = t + e \end{bmatrix}=\lim_{t \to 0} \left ( \ln(t + e)\right )^{1/t}=\\ &=e^{\displaystyle\lim_{t \to 0} \frac{\ln(\ln(t + e))}{t}}\\ \end{align}$$ Now the exponent becomes: $$\begin{align} \lim_{t \to 0} \frac{\ln(\ln(t + e))}{t} […]

finding the limit to $(3^n+5^n)^{\frac{1}{n}}$

I was wondering if one can find the limit to the sequence $\{a_n\}$, where: $$\large a_n=(3^n+5^n)^{\large\frac{1}{n}}$$ Without the use of a calculator.

Convergence and closed form of this infinite series?

If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using: $$A_n =\overbrace{\pi r^2}^\text{Area of circle}-\overbrace{\frac{1}{2} r^2 n \sin (\frac{2 \pi}{n})}^\text{Area of n-gon} =r^2(\pi-\frac{1}{2} n \sin (\frac{2 \pi}{n}))$$ I want to find the following sum (starting with […]

How to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$?

I need to prove that $\lim_{x\rightarrow \infty}\dfrac{x^2}{e^x}=0$.

How can we calculate the limit $\lim_{x \to +\infty} e^{-ax} \int_0^x e^{at}b(t)dt$?

I am looking at the following exercise: Let the (linear) differential equation $y’+ay=b(x)$ where $a>0, b$ continuous on $[0,+\infty)$ and $\lim_{x \to +\infty} b(x)=l \in \mathbb{R}$. Show that each solution of the differential equation goes to $\frac{l}{a}$ while $x \to +\infty$, i.e. if $\phi$ is any solution of the differential equation, show that $\lim_{x \to […]