$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$
36248
A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem
144422
How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$
49929
If $ \lim_{n \to \infty} (2 x_{n + 1} - x_{n}) = x $, then is it true that $ \lim_{n \to \infty} x_{n} = x $?
52632
$f\colon(0,\infty)\to \mathbb R$ be continuous ; $f(x)\le f(nx) , \forall n \in \mathbb N , \forall x >0$ , then $\lim_{x\to \infty} f(x)$ exists?
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
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
20764
Is it true that $\lim\limits_{x\to\infty}{x·P[X>x]}=0$?
27410
If the sequence satisfies the property lim$_{n\to \infty}(a_n-a_{n-2})=0$, prove that lim$_{n\to \infty}\frac{a_n-a_{n-1}}{n}=0$.
31836
Proof of $\lim_{n\to \infty} \sqrt[n]{n}=1$
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
44336
Is $\lim\limits_{x\to x_0}f'(x)=f'(x_0)$?
66441
Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
121217
How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
135700
Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists
153267
Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.
9694
If $(y_{2n}-y_n) \to 0$ then $\lim_{n\to \infty} y_n$ exists