For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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1answer
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Find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates

I am asked to find limits of integration for the interior region of sphere with center $(a,0,0)$ and radius $a$ using spherical coordinates. How can one do that? I know that one may use $$ x = r ...
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2answers
78 views
+100

Lipschitz-type estimate… True or false?

I have two parameters $\alpha,\varepsilon>0$ and the following difference: ...
3
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1answer
161 views
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Is there any solution to find a condition for $f(x)=a+bx^n+cx^2-dx>0$ to always hold true?

Okay, I am interested to know the criteria for a function to always hold $$f(x)=a+bx^n+cx^2-dx>0,$$ if it is given that $a, b, c>0$ and $n\in(-2,2)$ is some real number and $x>0$. My idea ...
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3answers
206 views
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Definition of $\frac{dy}{dx}$ where we cannot write $y=f(x)$

Informally, I can say that "for a small unit change in $x$, $\frac{dy}{dx}$ is the corresponding small change in $y$". This is however a bit vague and imprecise, which is why we have a formal ...
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1answer
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How to evaluate this limit about Bernoulli number?

First,we define $\displaystyle I_{1}\left ( x \right )=\frac{\sin x}{x}$, then $\displaystyle \lim_{x\rightarrow 0^+}I_{1}\left ( x \right )=1$, also we have \begin{align*} I_2\left ( x \right ...
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2answers
190 views
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A proviso in l'Hospital's rule

L'Hospital's Rule states that $$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ can be applied when: (1) $f$, $g$ are differentiable; (2) $g'(x) \neq 0$ for $x$ near $a$ (except ...