# list of common limits

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

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

• 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,  $\lim_{x\to\infty}\frac{x^{a}}{b^{x}}=0$  (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))

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

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

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

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

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

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

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

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

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

• $\tan\left(x+\frac{\pi}{2}\right)=\lim_{\xi\to\frac{\pi}{2}}\frac{\tan x+\tan% \xi}{1-\tan x\tan\xi}=\lim_{\xi\to\frac{\pi}{2}}\frac{\sec^{2}\xi}{-\tan x\sec% ^{2}\xi}=-\cot x$    (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 $z$,
$\lim_{n\to\infty}\frac{n^{n+1}}{z^{n+1}}\left(c+\frac{n}{z}\right)^{-(n+1)}=e^% {-cz}.$

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

PlanetMath,

## References

• 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
 Title list of common limits Canonical name ListOfCommonLimits Date of creation 2014-02-23 10:09:07 Last modified on 2014-02-23 10:09:07 Owner Wkbj79 (1863) Last modified by pahio (2872) Numerical id 29 Author Wkbj79 (2872) Entry type Feature Classification msc 26A06 Classification msc 26A03 Classification msc 26-00 Related topic LimitRulesOfFunctions Related topic ImproperLimits Related topic LimitExamples Related topic HalleysFormula