Question

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

Answer 1

l'Hopital's Rule

Indeterminate Form 1: `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.