$\int_0^\infty f(x) \; dx < \infty$ implies $\lim_{x \rightarrow \infty} x f(x) = 0$.
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$
40564
Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
151568
Limit Integral equal to a supremun: $\lim_{n\to \infty} \left[\int_a^b (f(x))^n dx\right]^{{1}/{n}}=\sup_{x\in [a,b]} f(x).$
28949
Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
19072
$\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
41434
Show that: $\lim \limits_{n \rightarrow+\infty} \int_{0}^{1}{f(x^n)dx}=f(0)$
301
Prove: $\int_0^\infty \sin (x^2) \, dx$ converges.
44129
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.
155475
$\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent
84969
When does $\lim_{n\rightarrow \infty} a_n = 0$ imply $\sum_{n = 0}^{\infty} a_n$ converges?
6620
If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
117246
Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$.
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
112285
Show that $\lim_{x\to\infty} f(x)=0 \implies \lim_{x\to\infty} f'(x)=0$
63889
Prove that if $u_n\rightarrow \infty$ then $\frac{1}{u_n} \rightarrow 0$
130161
Showing that if $\lim_{x\to\infty}xf(x)=l,$ then $\lim_{x\to\infty}f(x)=0.$