Limits of functions

Posted February 2nd, 2008 by Isoscel

We list some useful limits .

$\lim_{x\to 0}\frac{\sin x}{x}=1\:;\lim_{x\to 0}\frac{\sin x-x}{x^3}=-\frac{1}{6}$

$\lim_{x\to 0}\frac{\cos x-1}{x^2}=-\frac{1}{2}$

$\lim_{x\to 0}\frac{e^x-1}{x}=1\\:\lim_{x\to 0}\frac{e^x-1-x}{x^2}=\frac{1}{2}$

$\lim_{x\to 0}\frac{\ln (1+x)}{x}=1\:;\lim_{x\to 1}\frac{\ln x}{x-1}=1$

$\lim_{x\to 0}\frac{(1+x)^a-1}{x}=a$

$a>0\:\:\lim_{x\to 0}\frac{a^x-1}{x}=\ln a$

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