Articles of limits

Finding $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without L'Hopital and series expansion.

So we have to find $\lim\limits_{x\to 0} \frac{a^x-1}{x}$ without using any series expansions or the L’Hopital’s rule. I did it using both but I have no idea how to do it. I tried many substitutions but nothing worked. Please point me in the right direction.

Can $f(x,y) = |x|^y$ be be made continuous?

Can $f(x,y) = |x|^y$ be appropriately defined at (0,0) in order to be continous there . if we approach from path y=mx then f(x) becomes |x|^mx with x approaches to 0 .then by taking logs and we get infinity by infinity form . on solving with regular l’hop method ans comes cout to be e^-mx […]

taking the limit of $f(x)$, questions

How do I take the limit of the following functions? I had included some of my thoughts with them. $\lim_{x\to\infty}\dfrac{4x^3 – 2x + 1}{8x^3 + \sin(x^2) – x^{-1}}$; my thoughts: I am not sure about the bottom since there are the sine function and $-1$ power $\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}$; my thoughts: isn’t $e^x$ faster since $x^{x-1}$ is […]

How to find $\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$?

$$\lim_{(x,y)\rightarrow (0,0)} \frac{\sin(x\cdot y)}{x}$$ How can I find this limit?

Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$?

I’m trying to give at least some partial answers for one of my own questions (this one). There the following arose: $\hskip1.7in$ Why is $\lim_{x \to 0} {\rm li}(n^x)-{\rm li}(2^x)=\log\left(\frac{\log(n)}{\log(2)}\right)$? Expanding at $x=0$ doesn’t look reasonable to me since ${\rm li}(1)=-\infty$ and Wolfram only helps for concrete numbers, see here for example. Would a “$\infty-\infty$” […]

does $\lim_{N\to\infty}\frac{\sum_{i=1}^N a_i}{\sum_{i=1}^N b_i}$ converge to $\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$

Can this ever be the case? $$\lim\limits_{N\to\infty}\frac{\sum\limits_{i=1}^N a_i}{\sum\limits_{i=1}^N b_i} = \lim\limits_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$$ with $a_i>0$, $b_i>0$, $a_i<b_i$. As others pointed out simulations indicate convergence, but is there formal ground to it?

Harmonic number divided by n

This question already has an answer here: How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $? 8 answers

Proving $\lim_{x\to x_0}f(x)$ with epsilon delta definition

I’ve asked the question below before with no answer, but I would like to stress that this time it is not a homework question (and also that I’ve spent hours trying to come up with a solution). This is the question: Let f be a function defined around $x_o$. For every $\epsilon>0$ there’s some $\delta>0$ […]

$\lim_{x \to 0} \dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} = 1$ for any $f,g \in C^1$ that are tangent to $\text{id}$ at $0$ with some simple condition

Theorem For any real functions $f,g \in C^1$ such that $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$ and $x$ is strictly between $f(x)$ and $g(x)$ for any $x \ne 0$:   $f,g$ are invertible on some open neighbourhood of $0$   $\dfrac{f(x)-g(x)}{g^{-1}(x)-f^{-1}(x)} \to 1$ as $x \to 0$ Questions What is […]

Given $\lim\limits_{x\to 0} \frac {x(1+a\cos x)-b\sin x}{x^3}=1$, what is the value of $a+b$?

Given that $$\lim\limits_{x\to 0} \frac {x(1+a\cos x)-b\sin x}{x^3}=1$$ What is the value of $a+b$ My try $\lim\limits_{x\to 0} (\frac {x(1+a\cos x)}{x^3}-\frac {b\sin x}{x^3})=1$ $\lim\limits_{x\to 0} (\frac {(1+a×cos(x)}{x^2}-\frac {b}{x^2})=1$ $\lim\limits_{x\to 0} \frac {1+a\cos x-bx}{x^2}=1$ Apply L’Hôpital’s rule: $\lim\limits_{x\to 0} \frac {-a\cos x-b}{2x}=1$ Apply L’Hôpital’s rule again: $\lim\limits_{x\to 0} \frac {-a\sin x}{2}=1$ $\to$ $a=-2$ Is my approach […]