Some limits recur so frequently that they deserve a name of their own and memorisation: these are the standard limits.
Property — Standard limits
\lim_{x\to 0}\frac{\sin x}{x} &= 1 & \lim_{x\to 0}\frac{1-\cos x}{x^2} &= \frac{1}{2} \\[4pt] \lim_{x\to 0}\frac{e^x-1}{x} &= 1 & \lim_{x\to 0}\frac{\ln(1+x)}{x} &= 1 \\[4pt] \lim_{x\to 0}\frac{(1+x)^\alpha - 1}{x} &= \alpha & \lim_{x\to\pm\infty}\left(1+\frac{1}{x}\right)^x &= e \end{aligned}$$
Each of these limits appears, before the calculation, as an indeterminate form (the first five as , the last as ): their value must therefore be learnt and recognised in order to resolve similar cases quickly.
Links
Topics: Limits
Concepts: Limit · Standard limits
Skills: Using formulae