sources: https://planetmath.org/listofcommonlimits $\lim _{x \rightarrow a} c=c$ $\lim _{x \rightarrow a} x^{n}=a^{n}$ $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ [[Limit of sin(𝜃) over 𝜃 at 𝜃=0]] $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0$ $\lim _{x \rightarrow 0} \frac{\arcsin x}{x}=1$ $\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}=1$ [[Derivative of the natural exponential]] $\lim _{x \rightarrow 0} \frac{a^{x}-1}{x}=\ln a$ as long as $a>0$ [[Derivative of exponential with other bases]] $\lim _{x \rightarrow \infty} \frac{x^{a}}{b^{x}}=0$ $\lim _{x \rightarrow 0^{+}} x^{x}=1$ $\lim _{x \rightarrow 0^{+}} x \ln x=0$ $\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$ $\lim _{x \rightarrow \infty} x^{\frac{1}{x}}=1$ $\lim _{x \rightarrow \pm \infty}\left(1+\frac{1}{x}\right)^{x}=e$ $\lim _{x \rightarrow 0}(1+x)^{\frac{1}{x}}=e$ $\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{x}}=e$ $\lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-a^{2}}\right)=0$ $\lim _{x \rightarrow a} \frac{x-a}{x^{n}-a^{n}}=\frac{1}{n a^{n-1}}$ $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}}=\frac{1}{2}$ For $q>0, \lim _{x \rightarrow \infty} \frac{(\log x)^{p}}{x^{q}}=0$ $\lim _{n \rightarrow \infty} \frac{n^{n+1}}{z^{n+1}}\left(c+\frac{n}{z}\right)^{-(n+1)}=e^{-c z}$ $\lim _{n \rightarrow \infty}(\sqrt[n]{x}-1) n = \ln x$ $\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=1$