Following is a list of common limits used in elementary calculus:

• For any real numbers $a$ and $c$,  $\displaystyle\lim_{{x\to a}}c=c$.

• For any real numbers $a$ and $n$,  $\displaystyle\lim_{{x\to a}}x^{n}=a^{n}$  (proven here for $n$ a positive integer)

• $\displaystyle\lim_{{x\to 0}}\frac{\sin{x}}{x}=1$  (proven here)

• $\displaystyle\lim_{{x\to 0}}\frac{1-\cos{x}}{x}=0$  (proven here)

• $\displaystyle\lim_{{x\to 0}}\frac{\arcsin{x}}{x}=1$  (proven here)

• $\displaystyle\lim_{{x\to 0}}\frac{e^{x}-1}{x}=1$  (proven here)

• For $a>0$,  $\displaystyle\lim_{{x\to 0}}\frac{a^{x}-1}{x}=\ln a$ (proven here).

• For $b>1$ and $a$ any real number,  $\displaystyle\lim_{{x\to\infty}}\frac{x^{a}}{b^{x}}=0$  (proven here).

• $\displaystyle\lim_{{x\to 0^{+}}}x^{x}=1$  (proven here)

• $\displaystyle\lim_{{x\to 0^{+}}}x\ln{x}=0$  (proven here)

• $\displaystyle\lim_{{x\to\infty}}\frac{\ln{x}}{x}=0$  (proven here)

• $\displaystyle\lim_{{x\to\infty}}x^{\frac{1}{x}}=1$  (proven here)

• $\displaystyle\lim_{{x\to\pm\infty}}\left(1+\frac{1}{x}\right)^{x}=e$

• $\displaystyle\lim_{{x\to 0}}\left(1+x\right)^{\frac{1}{x}}=e$

• $\displaystyle\lim_{{x\to 0}}(1+\sin{x})^{\frac{1}{x}}=e$  (power of $e$, l’Hôpital’s rule)

• $\displaystyle\lim_{{x\to\infty}}(x-\sqrt{x^{2}-a^{2}})=0$  (proven here)

• For $a>0$ and $n$ a positive integer,  $\displaystyle\lim_{{x\to a}}\frac{x-a}{x^{n}-a^{n}}=\frac{1}{na^{{n-1}}}$.

• $\displaystyle\lim_{{x\to 0}}\frac{\tan x-\sin x}{x^{3}}=\frac{1}{2}$  (by l’Hôpital’s rule)

• For $q>0$, $\displaystyle\lim_{{x\to\infty}}\frac{(\log x)^{p}}{x^{q}}=0$

• $\displaystyle\tan\left(x+\frac{\pi}{2}\right)=\lim_{{\xi\to\frac{\pi}{2}}}\frac{\tan x+\tan\xi}{1-\tan x\tan\xi}=\lim_{{\xi\to\frac{\pi}{2}}}\frac{\sec^{2}\xi}{-\tan x\sec^{2}\xi}=-\cot x$    (by  l’Hôpital’s rule)
  That is, $\tan x\tan(x+\frac{\pi}{2})=-1$, which indicates orthogonality of the slopes represented by those functions.

• For a real or complex constant $c$ and a variable $z$,
  $\displaystyle\lim_{{n\to\infty}}\frac{n^{{n+1}}}{z^{{n+1}}}\left(c+\frac{n}{z}\right)^{{-(n+1)}}=e^{{-cz}}.$

• For $x$ real (or complex),  $\lim_{{n\to\infty}}n(\sqrt[n]{x}-1)=\log{x}$  (proven here for real $x$).

Feel free to add! Also, if the limit you decide to add is proven somewhere on PlanetMath, please provide a link. Thanks.