Limits of functions

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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