list of common limits

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

• For any real numbers $a$ and $c$,  $li{m}_{x\to a}c=c$.

• For any real numbers $a$ and $n$,  ${lim}_{x\to a}{x}^n=a^n$ (proven here (http://planetmath.org/ContinuityOfNaturalPower) for $n$ a positive integer)

• ${lim}_{x\to 0}\frac{\sin x}{x}=1$  (proven here (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0))

• ${lim}_{x\to 0}\frac{1-\cos x}{x}=0$  (proven here (http://planetmath.org/LimitOfDisplaystyleFrac1CosXxAsXApproaches0))

• ${lim}_{x\to 0}\frac{\arcsin x}{x}=1$  (proven here (http://planetmath.org/LimitExamples))

• ${lim}_{x\to 0}\frac{e^x-1}{x}=1$  (proven here (http://planetmath.org/DerivativeOfExponentialFunction))

• For $a>0$,  ${lim}_{x\to 0}\frac{a^x-1}{x}=\ln a$ (proven here (http://planetmath.org/LimitOfDisplaystyleFracax1xAsXApproaches0)).

• For $b>1$ and $a$ any real number,  ${lim}_{x\to \mathrm{\infty }}\frac{x^a}{b^x}=0$  (proven here (http://planetmath.org/GrowthOfExponentialFunction)).

• ${lim}_{x\to 0^{+}}{x}^x=1$  (proven here (http://planetmath.org/FunctionXx))

• ${lim}_{x\to 0^{+}}x\ln x=0$  (proven here (http://planetmath.org/GrowthOfExponentialFunction))

• ${lim}_{x\to \mathrm{\infty }}\frac{\ln x}{x}=0$  (proven here (http://planetmath.org/GrowthOfExponentialFunction))

• ${lim}_{x\to \mathrm{\infty }}{x}^{\frac{1}{x}}=1$  (proven here (http://planetmath.org/GrowthOfExponentialFunction))

• ${lim}_{x\to \pm \mathrm{\infty }}{\left(1+\frac{1}{x}\right)}^x=e$

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

• ${lim}_{x\to 0}{(1+\sin x)}^{\frac{1}{x}}=e$  (power of $e$, l’Hôpital’s rule (http://planetmath.org/LHpitalsRule))

• ${lim}_{x\to \mathrm{\infty }}(x-\sqrt{x^2-a^2})=0$  (proven here (http://planetmath.org/Hyperbola))

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

• ${lim}_{x\to 0}\frac{\tan x-\sin x}{x^3}=\frac{1}{2}$  (by l’Hôpital’s rule (http://planetmath.org/LHpitalsRule))

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

• $\tan \left(x+\frac{\pi }{2}\right)={lim}_{\xi \to \frac{\pi }{2}}\frac{\tan x+\tan \xi }{1-\tan x\tan \xi }={lim}_{\xi \to \frac{\pi }{2}}\frac{{\sec }^2\xi }{-\tan x{\sec }^2\xi }=-\cot x$   (by  l’Hôpital’s rule (http://planetmath.org/LHpitalsRule))
  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 $z$,
  ${lim}_{n\to \mathrm{\infty }}\frac{n^{n+1}}{z^{n+1}}{\left(c+\frac{n}{z}\right)}^{-(n+1)}=e^{-cz}.$

• For $x$ real (or complex),  ${lim}_{n\to \mathrm{\infty }}n(\sqrt[n]{x}-1)=\log x$  (proven here (http://planetmath.org/HalleysFormula) for real $x$).

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