if $f(x)\geq 0$ is uniformly continuous on $[1, \infty)$ and $\int_1^{\infty}f(x)dx$ converges, then $\lim_{x\rightarrow\infty}f(x)=0$
117486
Limit of uniformly continuous functions with convergent improper integrals.
13257
How to prove this problem about uniform continuity
20089
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
33337
$\int_{0}^{\infty} f(x) \,dx$ exists. Then $\lim_{x\rightarrow \infty} f(x) $ must exist and is $0$. A rigorous proof?
5229
Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.
48891
Show that: lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$
63889
Prove that if $u_n\rightarrow \infty$ then $\frac{1}{u_n} \rightarrow 0$
144530
is uniform convergent sequence leads to bounded function?
102076
Sequence of monotone functions converging to a continuous limit, is the convergence uniform?
10408
Uniformly convergence of $f_{n+1}(x)= \int^x_0 f_n(t)dt$
153267
Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
10428
Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
6620
If $\lim\limits_{x\rightarrow\infty} (f'(x)+f(x)) =L<\infty$, does $\lim\limits_{x\rightarrow\infty} f(x) $ exist?
56867
Convergence proof that involves integral
129551
Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?
46101
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
80622
Lebesgue integral uniform convergence