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Homelist of common limits

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# list of common limits

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$\tan\left(x+\frac{\pi}{2}\right)=\lim_{{\xi\to\frac{\pi}{2}}}\frac{\tan x+\tan% \xi}{1-\tan x\tan\xi}=\lim_{{\xi\to\frac{\pi}{2}}}\frac{\sec^{2}\xi}{-\tan x% \sec^{2}\xi}=-\cot x$ (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 $z$,

$\lim_{{n\to\infty}}\frac{n^{{n+1}}}{z^{{n+1}}}\left(c+\frac{n}{z}\right)^{{-(n% +1)}}=e^{{-cz}}.$ - •
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.

# 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

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## Comments

## Excellent entry

This is an excellent entry! Though it might not be complete (even if we take "common" to mean "quintessential"), already it shows that calculus can be about actual numbers people know and care about, like 1/2 and pi.

## Re: Excellent entry

Thanks, Lisa! Interessant, that the people first did not know what kind of limits one could list there.

Up to now, only Warren and I have added those limits -- we wish that also other people could increase the list =o)

Jussi

## What kinds of limits are common?

that the people first did not know what kind of limits one could list there ... we wish that also other people could increase the list =o)

They might still not know, they might have questions. Mine is: does Euler's gamma function expressed as a limit belong in here?

## Re: What kinds of limits are common?

I'd say the limit should be there, and the proof elsewhere. If the article becomes too long in the future I'm sure it can be broken into subsection and subentries. Until then, why not add away?

## Re: What kinds of limits are common?

I think this is an issue of intended audience. Presumably, the

more elementary identities are meant for the benefit of calculus

students seeing limits for the first time. Encountering yet

unfamiliar material such as gamma functions mixed in with the

elementary material can be disconcerting because it can makes one

unsure whether one has the necessary background to understand

even the elementary material. Therefore, I would suggest that

you add the limits involving gamma functions towards the end of

the list, after the more elementary ones. It might also be good to

split up the limits under headings such as "algebraic", "power",

"exponential", "trigonometric", "higher transcendents" for

ease of reference and further clarifying required background.

I think that if you include the gamma limits this way, the likely

result will be that beginners will view the advanced results as

something new, exciting, and mysterious to look ahead to rather

than as a confusing nuisance.