list of common limits

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

• For any real numbers $a$ and $c$ , $\ds \lim_{x\to a} c=c$ .
• For any real numbers $a$ and $n$ , $\ds \lim_{x\to a} x^n = a^n$ (proven here for $n$ a positive integer)
• $\ds \lim_{x\to 0} \frac{\sin{x}}{x}=1$ (proven here)
• $\ds \lim_{x\to 0} \frac{1-\cos{x}}{x}=0$ (proven here)
• $\ds \lim_{x\to 0} \frac{\arcsin{x}}{x}=1$ (proven here)
• $\ds \lim_{x\to 0} \frac{e^x-1}{x}=1$ (proven here)
• For $a>0$ , $\ds \lim_{x\to 0} \frac{a^x-1}{x}=\ln a$ (proven here).
• For $b>1$ and $a$ any real number, $\ds \lim_{x\to\infty}\frac{x^a}{b^x} = 0$ (proven here).
• $\ds \lim_{x\to 0^+} x^x = 1$ (proven here)
• $\ds \lim_{x\to 0^+} x\ln{x} = 0$ (proven here)
• $\ds \lim_{x\to\infty} \frac{\ln{x}}{x} = 0$ (proven here)
• $\ds \lim_{x\to\infty} x^\frac{1}{x} = 1$ (proven here)
• $\ds \lim_{x\to\pm\infty}\left(1+\frac{1}{x}\right)^x = e$
• $\ds \lim_{x\to 0}\left(1+x\right)^\frac{1}{x} = e$
• $\ds \lim_{x\to 0}(1+\sin{x})^\frac{1}{x} = e$ (power of $e$ , l'Hôpital's rule)
• $\ds \lim_{x\to\infty}(x-\sqrt{x^2-a^2}) = 0$ (proven here)
• For $a>0$ and $n$ a positive integer, $\ds \lim_{x\to a} \frac{x-a}{x^n-a^n}= \frac{1}{na^{n-1}}$ .
• $\ds \lim_{x\to 0} \frac{\tan x-\sin x}{x^3}= \frac{1}{2}$ (by l'Hôpital's rule)
• For $q > 0$ , $\ds \lim_{x \to \infty} \frac{(\log x)^p}{x^q} = 0$
• $\ds \tan\left(x+\frac{\pi}{2}\right)=\lim_{\xi\to\frac{\pi}{2}}\frac{\tan x+\tan\xi}{1-\tan x\tan\xi}= \lim_{\xi\to\frac{\pi}{2}}\frac{\sec^2\xi}{-\tan x\sec^2\xi}=-\cot x$ (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 $z$ ,
  $\ds \lim_{n\to\infty} \frac{n^{n+1}}{z^{n+1}}\left(c+\frac{n}{z}\right)^{-(n+1)}=e^{-cz}.$
• For $x$ real (or complex), $\lim_{n\to\infty} n(\sqrt[n]{x}-1)=\log{x}.$

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

Bibliography

1

Catherine Roberts & Ray McLenaghan, ``Continuous Mathematics'' in Standard Mathematical Tables and Formulae ed. Daniel Zwillinger. Boca Raton: CRC Press (1996): 333, 5.1 Differential Calculus