Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$
32063
if $f'(x)\rightarrow L$ as $ x \rightarrow \infty$, $-\infty \leq L \leq \infty $ then $ f(x)/x \rightarrow L $ as $x \rightarrow \infty$
104853
Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
83703
Show that $\int_{0}^{\infty}|\frac{\sin x}{x}|\,dx=\infty $
119602
$f:\mathbb{R} \to \mathbb{R}$ be differentiable and $\lim\limits_{x\to\infty}f'(x)=1$, is $f(x)$ unbounded?
81
$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$
122301
Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
120777
How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.
138930
What does $\lim\limits_{x \to \infty} f(x) = 1$ say about $\lim\limits_{x \to \infty} f'(x)$?
144422
How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$
132738
Showing $\lim_{n \to \infty} m(E_n) = 0$, assuming $f > 0$ a.e. and $\lim_{n \to \infty} \int_{E_n}f \,dm =0$
59162
Prove $\liminf\limits_{n\to\infty} F(x_n)\le F(x)$
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
104456
Show that $\lim_{n\to\infty}b_n=\frac{\sqrt{b^2-a^2}}{\arccos\frac{a}{b}}$
41037
Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
31685
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
30160
How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?