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

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 […]

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?

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.

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 […]

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 > […]

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

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 […]

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

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} […]

Intereting Posts

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Polynomials all of whose roots are rational
embdedding standard models of PA into nonstandard models
Relation between varieties in the sense of Serre's FAC and algebraic schemes
How do you use the BBP Formula to calculate the nth digit of π?
Geometry formulas, how to show identities.
generalized inverse of a matrix and convergence for singular matrix
How can LU factorization be used in non-square matrix?
Is there a classification of local rings with trivial group of units?
Continuous images of open sets are Borel?
Proving that $D_{12}\cong S_3 \times C_2$
Find a Continuous Function with Cantor Set Level Sets
$E|X-m|$ is minimised at $m$=median
Given $f ( x ) = x^n + a_{1} x_{n − 1} + ··· + a_{n − 1} x + a_n$, find the discriminant of the these polynomials
Let $G$ a group, with … Show that $G$ is a cyclic group.