# Questions tagged [calculus]

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

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### How to evaluate $\lim\limits_{h \to 0} \frac {3^h-1} {h}=\ln3$?

How is $$\lim_{h \to 0} \frac {3^h-1} {h}=\ln3$$ evaluated?
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### How do I find the minimum distance from a point to a graph?

Assume that I have a graph $$(C): y=f(x)$$ and a point $$A(x_0, y_0)$$ How do I find the minimum distance from point A to graph (C)?
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### How to calculate $\lim_{x \to 0}\left(\frac1{x} + \frac{\ln(1-x)}{x^2}\right)$?

How to calculate the following limit? $$\lim_{x \to 0}\left(\frac1{x} + \frac{\ln(1-x)}{x^2}\right)$$
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### Is there a name for this function?

this should be simple A polynomial could be defined as $$P_n (x) = \sum_{i=1}^{n} a_i x^{i-1}$$ Would the infinite-dimensional version of that F_l (x) = ...
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### How to find $\int\frac{\sin x}{x}dx$

How do I integrate $$\int\frac{\sin(x)}xdx$$? I tried using integration by parts, but it led me to nowhere. Please help.
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### convex inequality

why does it follow from convexity that $\sum^n_{i=1} 2^{k_i} \geq n2^{k/n}$, where $k_1 +...+k_n = k$ holds?
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### prove that $\lim\limits_{x\to 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}$

I'm trying to find a way to prove this: EDIT: without using LHopital theorem. $$\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}.$$ Honestly, I didn't come with any good idea. We know ...
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### More Computing Integrals

This particular problem has been giving me trouble, and while the math dept tutors did help a great deal, the resulting answer hasn't been accepted by the online homework submission website. Find the ...
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### Calculus one problem about substitution and area

I'm looking for a fun (not too many tedious calculations) calculus one problem that uses the concept that, after subsitution, you have two integrals of diffent functions with different limits, but ...
Can someone explain how to do this? area we're dealing with: $x^2 + y^2 + z^2 = a^2, z \geq 0$ I'm aware that the answer is: $x = a \sin(\phi) \cos(\theta)$ $y = a \sin(\phi) \sin(\theta)$ $z = ... 4answers 1k views ### Proof that this limit equals$e^a$Can someone please explain to me why the following identity is true? $$\lim_{x \to \infty}\left(1 + \frac{a}{x} \right)^x = e^a$$ (I'll make a notation$L$that is equal to the limit above.) One '... 2answers 325 views ### Is there a difference between these integral notations? I've come across these two notations for calculating an indefinite integral but I'm not sure whether or not they are equal:$f(x)dx\int f(x)dx$When calculating the indefinite integral, the first ... 1answer 446 views ### How to evaluate$\int \frac{\cos(x) - 1}{x^2}\mathrm dx\$?
would like a hint with the integral $$\int \frac{\cos(x) - 1}{x^2}\mathrm dx$$Thanks