Step By Step Calculus » 13.3 - L'Hospital's Rule

show_hiddenExpand All [+]
Synopsis
There is a method involving derivatives that is very useful in solving limits of an indeterminate form. This method is called pital’s ruleL’Hópital’s rule:
L’Hópital’s rule
If the limit \displaystyle \lim_{t\to a}\frac{y(t)}{x(t)}\displaystyle \lim_{t\to a}\frac{y(t)}{x(t)} is of indeterminate form \displaystyle \frac{0}{0}\displaystyle \frac{0}{0} or \displaystyle \frac{\infty}{\infty}\displaystyle \frac{\infty}{\infty}, then
\displaystyle {\lim_{t\rightarrow a}\frac{y(t)}{x(t)} =\lim\limits_{t\rightarrow a}\frac{y'(t)}{x'(t)},}\displaystyle {\lim_{t\rightarrow a}\frac{y(t)}{x(t)} =\lim\limits_{t\rightarrow a}\frac{y'(t)}{x'(t)},}
provided x'(t), y'(t)x'(t), y'(t) exists with x'(t)\ne 0x'(t)\ne 0 on (a-\delta, a)\cup(a,a+\delta)(a-\delta, a)\cup(a,a+\delta) for some \delta>0\delta>0.
L’Hópital’s rule is immensely useful because it is possible to convert other indeterminate forms into the \dfrac{\infty}{\infty}\dfrac{\infty}{\infty} or \dfrac{0}{0}\dfrac{0}{0} indeterminate forms (for which this rule is applicable) through algebra or application of \ln\ln.