Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $
61808
Putnam 1967 problem, integration
153267
Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
140574
Find $\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx$
73812
Does $\lim \limits_ {n\to \infty}\ \frac{n}{n+1}$ equal $1$?
104853
Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
138973
How prove this limit $\lim\limits_{n\rightarrow \infty} \frac{f_n}{f_{n+1}}=a$ given two other limits related to $f_n$
44336
Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?
129042
Prove that $\lim_{a \to 0^{+}} \int_{0}^{a} \frac{1}{\sqrt{\cos(x)-\cos(a)}} \;dx=\frac{\pi}{\sqrt{2}}$
57600
Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $
48891
Show that: lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
117246
Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$.
113141
Prove that $\lim_{n \to \infty}\bigg[\int_0^1f(t)^n \text{dt}\bigg]^{1/n}=M$
119145
$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$
155333
Prove that $\lim_{x\to\infty}\frac{f(x)}x=\lim_{x\to\infty}f'(x)$
144273
Find $\lim_{n\to\infty} \frac{1}{n} \int_0^\infty f(x) e^{-a_n x} dx$
67645
Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if $a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$
126736
Show that $\lim_{\epsilon\to0^{+}}\frac{1}{2\pi i}\int_{\gamma_\epsilon}\frac{f(z)}{z-a} \, dz=f(a)$