User Mark Viola

70 votes

Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$ if $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$

answered Apr 18, 2018 at 17:55

66 votes

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

answered Mar 31, 2015 at 17:28

56 votes

Limit of $n$th root of $n$!

answered Feb 24, 2017 at 23:58

54 votes

Adding two polar vectors

answered Jul 18, 2015 at 21:07

48 votes

How to prove that $\log(x)<x$ when $x>1$?

answered Dec 27, 2015 at 6:11

47 votes

Accepted

Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$

answered Nov 5, 2016 at 2:05

42 votes

The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)

answered Feb 23, 2017 at 4:22

38 votes

Accepted

What is the primitive function of $\int 1/(x^{2n} +1)dx$?

answered Jul 8, 2015 at 22:33

36 votes

Accepted

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

answered Aug 6, 2015 at 3:18

35 votes

Accepted

Improper integral of sin(x)/x from zero to infinity

answered Jun 14, 2017 at 16:22

34 votes

Accepted

Partial derivatives inverse question

answered Apr 15, 2016 at 19:53

34 votes

Accepted

How to determine whether this function is differentiable at a point?

answered Oct 4, 2015 at 23:34

32 votes

Stirling's formula: proof?

answered Mar 15, 2017 at 23:34

31 votes

Counterexample to Leibniz criterion for alternating series

answered Feb 1, 2018 at 22:17

31 votes

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

answered Dec 6, 2016 at 18:47

30 votes

Accepted

Exact Sum of Series

answered Dec 2, 2015 at 20:32

29 votes

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$

answered Oct 2, 2015 at 17:49

29 votes

Accepted

Function that looks a lot like exponential, but isn't

answered Jul 19, 2015 at 16:01

26 votes

Accepted

How to evaluate $\int_0^1\int_0^1 \frac{1}{1-xy} \, dy \, dx$ to prove $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$.

answered Aug 7, 2016 at 15:13

26 votes

Fourier transform of Bessel functions

answered Jul 14, 2015 at 3:19

25 votes

Accepted

How to obtain the Laurent expansion of gamma function around $z=0$?

answered Aug 4, 2018 at 15:15

25 votes

Find the limit of $\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$ without L'Hospital's rule

answered Dec 21, 2017 at 16:48

24 votes

Accepted

Fundamental solution for Helmholtz equation in higher dimensions

answered Apr 28, 2017 at 6:21

23 votes

Are there any limit questions which are easier to solve using methods other than l'Hopital's Rule?

answered Nov 17, 2015 at 23:59

23 votes

Accepted

Complex keyhole contour integral

answered Apr 22, 2017 at 3:52

23 votes

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

answered Jun 29, 2015 at 16:46

22 votes

Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

answered Oct 19, 2017 at 18:21

22 votes

Accepted

Proving $\int_0^{\infty} \frac{\sin^3(x)}{x^2} dx = \frac{3\ln(3)}{4} $

answered Apr 30, 2020 at 3:29

21 votes

Accepted

Convergence/Divergence of some series

answered Jun 15, 2017 at 2:29

21 votes

Accepted

Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

answered Aug 29, 2015 at 17:01

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5,282 Answers

Prove that $\lim_{x\rightarrow 0}\frac{f(x^2)-f(0)}{x}=0$ if $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x=0$

How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?

Limit of $n$th root of $n$!

Adding two polar vectors

How to prove that $\log(x)<x$ when $x>1$?

Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$

The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)

What is the primitive function of $\int 1/(x^{2n} +1)dx$?

How to find an approximation to $1 - \left( \frac{13999}{14000}\right )^{14000}$?

Improper integral of sin(x)/x from zero to infinity

Partial derivatives inverse question

How to determine whether this function is differentiable at a point?

Stirling's formula: proof?

Counterexample to Leibniz criterion for alternating series

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

Exact Sum of Series

How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$

Function that looks a lot like exponential, but isn't

How to evaluate $\int_0^1\int_0^1 \frac{1}{1-xy} \, dy \, dx$ to prove $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$.

Fourier transform of Bessel functions

How to obtain the Laurent expansion of gamma function around $z=0$?

Find the limit of $\lim_{x\to0}{\frac{\ln(1+e^x)-\ln2}{x}}$ without L'Hospital's rule

Fundamental solution for Helmholtz equation in higher dimensions

Are there any limit questions which are easier to solve using methods other than l'Hopital's Rule?

Complex keyhole contour integral

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

Proving $\int_0^{\infty} \frac{\sin^3(x)}{x^2} dx = \frac{3\ln(3)}{4} $

Convergence/Divergence of some series

Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$