Articles of limits

What does $\lim\limits_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$ evaluate to?

What does $$\lim_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$$ evaluate to? This very likely needs substitution.

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow \infty} \binom{n}{k}\frac{\binom{\binom{n-k}{2}}{N_c}}{\binom{\binom{n}{2}}{N_c}}=\frac{e^{-2kc}}{k!}.$$ It feels like using Stirling’s approximation would help but I can’t quite figure out how… I ask this question because I am currently trying […]

Show $\lim\limits_{n\to\infty} \frac{2n^2-3}{3n^ 2+2n-1}=\frac23$ Using Formal Definition of Limit

I want to show that $a_n=\frac{2n^2-3}{3n^ 2+2n-1}$ is convergent. So I did the following: \begin{align*} \left|a_n-\frac23\right|&=\left|\frac{2n^2-3}{3n^ 2+2n-1}-\frac23\right|\\ &=\left|\frac{-4n-7}{3(3n^2+2n-1)}\right| \\ &<\left|\frac{4n}{3n^2}\right|\tag{$\ast$}\\ &<\left|\frac4n\right|\\ &<\frac4N\\\ \end{align*} But I am not one hundred percent sure about ($\ast$) because $|-4n-7|=|4n+7|\not<4n$. Can somebody please explain my error in reasoning?

How can one show that $ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$?

Let $f:[0, 1] \rightarrow \mathbb{R}$ a continuous function. If $a>0$, show that: $$ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$$ Tried using Riemann sum, but did not succeed.

Selection of $b_n$ in Limit Comparison Test for checking convergence of a series

I wonder what the correct criterion for selection of $b_n$ in Limit Comparison Test for checking convergence of a series. Any hint to online material will be highly appreciated. The selection of $b_n$ in first series is easy but in other three is tricky. Is there any universal criterion for selection of $b_n$? Thanks in […]

Calculating derivative by definition vs not by definition

I’m not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. The following two examples confused me: $$ g(x) = \begin{cases} x^2\cdot \sin(\frac {1}{x}) & x \neq 0 \\ 0 & x=0 \end{cases} $$ $$ f(x) = \begin{cases} e^{\frac {-1}{x}} & x > […]

$\lim_{x \to 2} \frac{x^{2n}-4^n}{x^2-3x+2}$

Calculate the limit $$\lim_{x \to 2} \frac{x^{2n}-4^n}{x^2-3x+2}$$ I tried to use $$\lim_{x \to 2} \frac{(x^2)^n-4^n}{x^2-3x+2}$$ but i can’t find anything special

A representation of Dirac-$\delta$

Prove that $$g_\epsilon (x)=\lim_{\epsilon \to 0} \frac1 \epsilon \frac1 \pi e^{-x^2/\epsilon^2}$$ is a Dirac-$\delta$ function. This is a homework question I’m stuck with. I’m probably missing a very simple point, and can’t seem to figure it out. Any help to prompt me in the right direction would be much appreciated. What I’ve done so far […]

$\lim_{n \to \infty} \mid a_n + 3(\frac{n-2}{n})^n \mid^{\frac1n} = \frac35$. Then find $\lim_{n \to \infty} a_n$.

This question already has an answer here: What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$? 3 answers

Limit of integral – part 2

Inspired by the recent post “Limit of integral”, I propose the following problem (hoping it will not turn out to be too easy). Suppose that $g:[0,1] \times [0,1] \to {\bf R}$ is continuous in both variables separately. Is it true that, for all $x_0 \in [0,1]$, $$ \lim \limits_{x \to x_0 } \int_0^1 {g(x,y)\,{\rm d}y} […]