# Tagged Questions

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### L'Hopital's rule question

How do you evaluate the limit $$\lim_{x \to 0^+} \left(x^{\tan x} + x^2 \frac{\sin(1/x)}{\tan x}\right)?$$ The answer is 1. (the first part is 1 and the second part is 0.) Thanks in advance!
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### Evaluate $\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x}$ without L'Hopital

I need help finding the the following limit: $$\lim_{x\rightarrow 0} \frac{\sin x}{x + \tan x}$$ I tried to simplify to: $$\lim_{x\rightarrow 0} \frac{\sin x \cos x}{x\cos x+\sin x}$$ but I ...
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### How to compute $\lim_{n \to \infty} \left( 1-\frac{2t}{n^2} \right)^{-n/2}$?

$$\lim_{n \to \infty} \left( 1-\frac{2t}{n^2} \right)^{-n/2}$$ How can I find the limit above? I am bit confused because of the square in the denominator. If it was $n$ instead, then the limit ...
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### Evaluate the limit $\lim_{x\to\infty}\frac{\ln(5e^{3x})}{\ln(3e^{5x})}$

I am trying to evaluate the following limit: $$\lim_{x\to\infty}\frac{\ln(5e^{3x})}{\ln(3e^{5x})}$$ So, I did the following: \begin{align} \require{cancel} ...
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### Evaluate $\lim_{x\to\infty} (1+\frac{2}{x})^{5x}$ without L'Hopital

I'm trying to evaluate the following limit $$\lim_{x\to\infty} \left(1+\frac{2}{x}\right)^{5x}$$ I recognize a part of this limit because it resembles the limit for $e$ but I don't know anything ...
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### Why is true? $\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}$

$$\begin{array}{l}a,b > 0\\\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}\\\end{array}$$ I asked already a similar question, but I'm still not sure what ...
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### Limit involving $\frac{0}{0}$ and $\sin 2x$

I'm currently working with the limits: $$\frac{\sin(2x)}{e^x-1}$$ for $x\to 0$ and $x\to \infty$ how can this be solved? I tried to reformulate $\sin(2x)$ to $2\sin x\cos x$ Using L'Hõpital's ...
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### find the limit of $\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$

$$a,b \gt 0$$ $$\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$$ So, I know that if x is $x \in \mathbb{Z}$ then the limit is $a\over [b]$ I couldn't figure out the ...
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### Prove: if every monotonic subsequence of $x_n$ converges to $x$ then $x_n$ converges to $x$

Prove: if every monotonic subsequence of $x_n$ converges to $x$ then $x_n$ converges to $x$ At first, it looked to me like an easy to solve question, but actually I'm kinda stuck. Can you give me a ...
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### An example of a function over the reals that does not have a limit as x go to 0 but does when in substraction

Give an example of a function $f:\mathbb R \to \mathbb R$ that the limit $\displaystyle\lim_{x\to 0}(f(x)-f(2x))$ exists but $\displaystyle\lim_{x\to 0}f(x)$ does not exists. I tried a few trig ...
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### Trying to prove a limit of a simple multi case function

Full disclosure: This is my attempt at solving a homework assignment Another full disclosure: This is my first time using LaTeX, so pardon any errors We were asked to prove / disprove the existance ...
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### Showing a sequence convergence

Let $a_1,a_2>0$ and $a_{n+1}=\cfrac{2}{a_{n-1}+a_{n}}(n\ge2)$, How to prove $a_n$ is convergent?
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### Finding the limit of $\mathop {\lim }\limits_{x \to 0} \frac{x}{{\ln (5x + 1)}}$

$$\lim_{x \to 0^+} \frac{x}{\ln (5x + 1)} = {1 \over 5}$$ First, What I tried to do is dividing by $x$, but it didn't work out. By the way, It's a common approach to find a limit. Why is it failing ...
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### Prove the limit $\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$

Prove the following limit: $$\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$$ I can use limits arithmatic, squeezing principle, "well-known" limits etc.. We didn't learn Lopital law so I ...
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### what is the limit of $\displaystyle \lim_{x\to 0} \frac{x}{a}\cdot \lfloor{\frac{b}{x}\rfloor}$

find the limit of: $\displaystyle\lim_{x\to 0} \frac{x}{a}\cdot \lfloor{\frac{b}{x}\rfloor}$ $a,b>0$ i know that: $\lfloor\frac{b}{x}\rfloor\le \frac{b}{x} + 1$; and ...
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### $\displaystyle L=\lim_{x\to 4}\left(4-\dfrac x2\right)$: find $\delta\gt0$ such that $\left|f(x)-L\right|\lt0.01$ whenever $0\lt|x-4|\lt\delta$.

Find $L$, where $\displaystyle L=\lim_{x\to 4}\left(4-\dfrac x2\right)$ Then find $\delta\gt0$ such that $\left|f(x)-L\right|\lt0.01$ whenever $0\lt|x-4|\lt\delta$. $L$ is easy to find, but ...
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### How should I define the limit definition of a derivative using negative numbers?

Typically the derivative is defined at a point $x$, assuming it is differentiable at it, by $$\lim_{n \rightarrow \infty} \frac{f(x + \frac{1}{n}) - f(x)}{\frac{1}{n}}$$ ...
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### prooving the limit of a function in epsilon,delta, and sequences

Hi I need to prove the following limit of a function: I need to show the proof in two ways, one in epsilon and delta and the other with the sequence the diverge to infinity. I'm having difficulties ...
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### How do I solve this limit?

$$\lim _{x \rightarrow 0} \left(\frac{ \sin x}{x}\right)^{1/x}$$ I have spent an hour on the above limit and have no work to show. I tried using L'Hopital's Rule, but just kept going around in ...
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### What is $\lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2}$?

I have limit: $$\lim_{(x,y)\to(2,2)}\frac{x^4-y^4}{x^2 - y^2}$$ Why is the result $8$ ?
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### Finding the limit: $\displaystyle \lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$

Can anyone tell me how to properly solve this limit? $\displaystyle \lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$ I know the answer is positive infinity, and I would know how to do the problem if $x$ was ...
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### Help clarify the limit of $e$ when there is an exponent

I want to compute: $$\lim_{n\to \infty} \left(1-\frac{\lambda}{n}\right)^n$$ I know that: $$e = \lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x$$ So, I let $x=-\frac{n}{\lambda}$ and get: ...
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### When $x$ goes to $0$ , what happens to $\sin\left(\frac{1}{x}\right)$ and $\cos\left(\frac{1}{x}\right)$?

When $x$ approaches $0$, do $\sin\frac{1}{x}$ and $\cos\frac{1}{x}$ converge or diverge? How do you show this?
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### Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don't understand how to approach this question
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### Cesaro means and equivalent sequences

Let $(u_n)$ be a sequence of complex numbers that converges in mean (Cesaro convergence). Let $(v_n)$ be a sequence such that $v_n\sim u_n$. Does the sequence $(v_n)$ converge in mean? Here is ...
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### AP Calculus Behavior

Which of the following describes the behavior of $y=\sqrt[3]{x+2}$ at x = -2 a) Differentiable b) Corner c) Cusp d) Vertical Tangent e) Discontinuity Which one is it and why? I know it is Vertical ...
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### Formal Definition of Limit and Proofs

I'm having trouble understanding the formal definition of a limit... Let $f(x)$ be defined on an open interval about $x_0$, except possibly at $x_0$ itself. We say that the limit of $f(x)$ as ...
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### Prove that $\left(a_{n}\right)_{n=1}^{\infty}$ converges when $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ for $0<q<1$

I'm stuck on a homework question, and could really use some help. Here is said question: "Assume that for every $n$ the following occurs: $|a_{n+1}-a_{n}|<q|a_{n}-a_{n-1}|$ when $0<q<1$ ...
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### Prove that $f(x)=\left\vert\,x\,\right\vert - \sin\left(\left\vert\, x\,\right\vert\right)$ is differentiable or discontinuous or has no derivative

I have no idea what to do with question number 9, and question number I used $\operatorname{sign}$ for the absolute but I don't know if I used right or wrong, more detail in the picture:
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### Prove that the sequence $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$ converges using Cauchy

I need some help with a homework question i'm having difficulty with. Here is the question: "Use the definition of cauchy sequence to prove that the series ...
### If $\lim_{x\rightarrow a} f(x) = 0$, and $\lim_{x\rightarrow a} g(x) = \infty$, show that $\lim_{x\rightarrow a} f(x)^{g(x)} = 0$
For some reason, the answer I get is $1$. \begin{align*} \lim_{x\rightarrow a} f(x)^{g(x)} &=\lim_{x\rightarrow a} g(x) \ln[f(x)]\\ &=\lim_{x\rightarrow a} \ln[f(x)] / (1/ {g(x)}) ...