What is L'hospital's Rule?

Question

What is  L'hospital's Rule?

Emilia Aguilar · Accepted Answer

undefined L'Hôpital's Rule is a mathematical theorem used for evaluating limits of functions that are in an indeterminate form, such as $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $. The rule states that if the limit of the quotient of two functions $ f(x) $ and $ g(x) $ as $ x $ approaches a certain value $ c $ is of the form $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $, then the limit of their derivatives $ f'(x) $ and $ g'(x) $ as $ x $ approaches $ c $ will be the same as the limit of the original quotient.

In other words, if $ \lim_{{x \to c}} \frac{f(x)}{g(x)} $ is of the form $ \frac{0}{0} $ or $ \frac{\infty}{\infty} $, then:

$$ \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} $$

This rule is particularly useful in calculus when dealing with limits involving functions that result in these indeterminate forms, allowing for easier evaluation of such limits.

Charles Arocha · Answer

l'Hopital's Rule If #{(lim_{x to a}f(x)=0 and lim_{x to a}g(x)=0), (or),
(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)}#. Example 1 (0/0) #lim_{x to 0}{sinx}/x=lim_{x to 0}{cosx}/1={cos(0)}/1=1/1=1# Example 2 (#infty#/#infty#) #lim_{x to infty}{x}/{e^x}=lim_{infty}{1}/{e^x}=1/{e^{infty}}={1}/{infty}=0# I hope that this was helpful.