# Tagged Questions

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### Left hand and right hand limits

what are the answers of these two limits? $$\lim_{x\to h^+}c_1\cos(x-a)+c_2\sin (x-a)$$ $$\lim_{x\to h^-}c_1\cos(x-a)+c_2\sin (x-a)$$ Thanks.
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### Limit of the supremum

Let $f:\mathbb{R}\to\mathbb{R}^n$ be an absolutely continuous function that $|f(t)|$ is nonincreasing, as $t\to\infty$. How can I compute \begin{equation*} ...
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### Find all points where $f(x)$ fails to be differentiable. Justify your answer

Find all points where $f(x)$ fails to be differentiable. Justify your answer $$f(x) = |x| - 1$$ I am confused with continuity of it and cannot turn it into piecewise function and finding the limit ...
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### Evaluating limits by subsituting special sequences, justification for that

Sometimes I saw people using transformations like $$\lim_{x\to 0} f(x) = \lim_{n\to \infty} f(\frac{1}{n})$$ or $$\lim_{x \to p} f(x) = \lim_{n \to \infty} f(x + \frac{1}{n}). \quad (*)$$ I know ...
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### Show $g(x)=\sqrt{x}$ is continuous at x=4

I just really need to make sure that I am understanding the process for doing these. Scratch work: We have $|\sqrt{x}-\sqrt{4}| = |\sqrt{x}-2|= |\frac{x-4}{\sqrt{x+2}}|= \frac{|x-4|}{|\sqrt{x}+2|}$. ...
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### The limit of arccos(x) is the arccos(x) of the limit?

Recently I came across a proof using the apparant fact that $$\lim_{x \rightarrow a} \cos^{-1}(x)=\cos^{-1}(\lim_{x \rightarrow a} x)$$ with justification: because arccos(x) is a continuous function. ...
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### Multi-variable continuity piecewise problem

I've worked on this for about 3 hours and I can't seem to get anywhere with it. I tried using java code to return the solution but one that met the criteria was not found. Find the value of $c$ and ...
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### Explain this consequence of continuity

A consequence of continuity is the following fact: if $f(x)$ is continuous at $x=b$ and $\lim\limits_{x \to a} g(x)=b$, then, $\lim\limits_{x \to a} f(g(x)) = f(\lim\limits_{x \to a}g(x))$ with ...
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### Limit with integral or is this function continuous?

Hello I need to show one identity and one limit. I am having problems with it. notation: $x_i$ is i-th coordinate of $x$ $B(x,r)$ ball with center $x$ and radius $r$ $S(x,r)$ sphere with center ...
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### $f:\mathbb{R}\to\mathbb{R}$ is continuous and $\int_{0}^{\infty} f(x)dx$ exists

$f:\mathbb{R}\to\mathbb{R}$ is continuous and $\int_{0}^{\infty} f(x)dx$ exists could any one tell me which of the following statements are correct? \$1. \text{if } \lim_{x\to\infty} f(x) \text{ ...