How do you use L'hospital's rule to find the limit?

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Answer 1 · 2014-10-13T08:38:26

l'Hopital's Rule

Indeterminate Form 1: $0$ $0$ / $0$ $0$

If $lim_{x to a}f(x)=0$ and $lim_{x to a}g(x)=0$ ,

then $lim_{x to a}{f(x)}/{g(x)}=\lim_{x to a}{f'(x)}/{g'(x)}$ .

ex.) $lim_{x to 0}{sinx}/{x}$

by differentiating the numerator and the denominator separately,

$=lim_{x to 0}{cosx}/{1}=cos(0)=1$

Indeterminate Form 2: $infty$ / $infty$

If $lim_{x to a}f(x)=pm infty$ and $lim_{x to a}g(x)=pm infty$ ,

then $lim_{x to a}{f(x)}/{g(x)}=\lim_{x to a}{f'(x)}/{g'(x)}$ .

ex.) $lim_{x to infty}{x}/{e^x}$

by differentiating the numerator and the denominator,

$=lim_{x to infty}{1}/{e^x}=1/infty=0$

Note: There are other indeterminate forms which can be turned into one of the above forms.

I hope that this was helpful.