Questions tagged [limits]

Question 1

I am currently taking a real analysis class, and we've learned some notions from topology (nothing deep just key concepts). While I was proving a theorem, I came across something that really confused ...

Question 2

I recently came across a surprising (to me) limit that I wanted to try to understand a bit better. The limit is: $$ \lim_{N \to \infty} | N \cdot ( \ln(2) - (1 - 1/2 + 1/3 + \ldots + (-1)^{N-1} / N)| ...

Question 3

Let $\alpha > 0$ and consider the function $$ f(x,y) = \begin{cases} \dfrac{|\sin(xy) - xy|^{\alpha}}{(x^{2} + y^{2})^{3}} & \text{if } (x,y) \neq (0,0), \\[2mm] 0 & \text{if } (x,y) = (0,...

Question 4

If a real or complex function $f$ is undefined at a point $a$, but is sufficiently well-behaved near that point, we can find a sort of "average value" or "finite part" of $f$ at $a$...

Question 5

I'm trying to prove the following limit without using Taylor expansion or L'Hopital rule: $$\lim_{x \to 0} \frac{e^x - e^{\sin(x)}}{x-\sin(x)}=1$$ But I don't know how to do it. I tried with some ...

Question 6

As something adjacent to the birthday problem, I found that $\frac1{\ln{365}-\ln{364}}$ is very nearly $364$ or $365$, but much closer to $364.5$. This leads to $$\ln{\frac{x+0.5}{x-0.5}}\approx \...

Question 7

I'm doing a proof on Graph theory on a colorability problem of a connected graph. My proof method involves decomposing numbers into the sum of smaller numbers according to specific rules. In ...

Question 8

I have $\lim_{x \to 0} \frac{x^2 - \ln^2(1+x)}{x^3}$ and the value of this limit is 1. I tried to solve: $$\begin{align}\lim_{x \to 0} \frac{x^2 - \ln^2(1+x)}{x^3} &= \lim_{x \to 0} \frac{(x - \ln(...

Question 9

$$\lim_{x\to0}\frac{\sqrt[3]{1+mx}-1}{x}$$

Question 10

Let $X$ be a first countable topological space. Let $\phi:D\to X$ be a net such that $D$ has a countable cofinal subset $C$. Without loss of generality, we can think of $C$ to be indexed by $\mathbb N$...

Question 11

NOTE- The source of question is Advanced Calculus on real axis which doesn't cover $\textsf{Lebesgue integral}$ and $\textsf{Measure Theory}$ so therefore an approach by those wouldn't be of much ...

Question 12

So I was looking over at this limit $$\lim_{x \to -2}\frac{bx^{2} + 15x +15 + b}{x^{2}+x-2}$$ Suppose the limit exists and we're suppose find the corresponding value of $b$. My understanding is that ...

Question 13

I know that the limit can be proven using the standard $\varepsilon$-$\delta$ definition of limits but is my proof valid? If not, what is it lacking and how is it flawed? Below is my approach to the ...

Question 14

Let $\alpha > 0$ and $f : \mathbb{R}^{2} \to \mathbb{R}$ be the function defined by $$ f(x, y) = \begin{cases} \dfrac{\log(1 + |xy|^\alpha)}{x^2 + y^4}, & \text{if } (x, y) \neq (0, 0), \\ 0, ...

Question 15

I know it can be solved with Squeeze's theorem, but I want to verify that this more conventional method might also be valid. $$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...

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Questions tagged [limits]

Definition of one sided limit

Why do the Genocchi numbers appear in this series?

Analysis of continuity for a two-variable function with a power parameter $\alpha$

Is there a name for the limit $\lim_{\epsilon \to 0} \frac{f(a - \epsilon) + f(a + \epsilon)}{2}$?

How to solve $\lim_{x \to 0} \frac{e^x - e^{\sin(x)}}{x-\sin(x)}$

Why does $\left(1+\frac1{x-0.5}\right)^x$ converge to $e$ quicker than $\left(1+\frac1x\right)^x\approx e$?

Formulas to find out the critical limit of decomposing numbers

Evaluate $\lim_{x \to 0} \frac{x^2 - \ln^2(1+x)}{x^3}$ without using L'hopital's rule , Series Expansion or Integrals

Computing the limit $\lim_{x\to0}\frac{\sqrt[3]{1+mx}-1}{x}$ [closed]

In a first countable space, cluster points of a net with a countable cofinal subset are necessarily limits of sequential subnets?

Let$\ \ f:[0,1]\to \mathbb{R}$ be a continuous function. Prove $\ \ \lim_{\lambda\to\infty}\int_0^1 f(x)\sin(\lambda x)\,dx = 0$. [duplicate]

Question regarding the usage of L'Hopital's rule.

Is this a valid proof that $\displaystyle \lim_{x\rightarrow\infty}\frac{1}{x^2}=0$? [closed]

Continuity and differentiability at the origin of a logarithmic function with a parameter $\alpha$

How to solve $ \lim\limits_{x\to+\infty} \!\!\left(\! \frac{x^{2}+3}{3x^{2}+1}\! \right)^{\!x^{2}}\!\!\!=0\;?$

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