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#

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.