Prove that $\lim_{\varepsilon\to 0^+}\int_{a\varepsilon}^{b\varepsilon}\dfrac{f(x)}{x}\ \text{d}x=f(0)\ln(b/a)$.
132747
How can one show that $ f(0)\ln(\frac{b}{a})=\lim_{\epsilon\rightarrow 0}\int_{\epsilon a}^{\epsilon b} \frac{f(x)}{x}dx$?
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
157577
Prove that $\lim_{x\to y} \frac{d(x, F )}{|x−y|} = 0$ for a.e. $y \in F$.
50673
Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
53688
How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$?
66441
Prove that $|x|<1$ implies that $\lim_{n \to \infty} x^n=0$
74583
Prove that $f'(0)=L$.
69348
Prove that $\int \limits _{0}^{\infty}\frac{e^{-2x}-e^{-ax}}{x}\text{d}x$ converges for any $a>0$
49408
Proof of the derivative of ln(x)
68818
$\lim_{h\to 0}\int \frac{f(x+h)-f(x)}{h}\,\mathrm{d}x = f(b)-f(a)$
150308
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
53236
Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$
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
67622
Proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$
85269
Proof that limit of (lnx)/x = 0 without LHopital
27511
Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$
67645
Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if $a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$