What is L'Hospital's rule?
0:45 // What does L'Hospital's rule do?
2:36 // How many times can you apply L'Hospital's rule?
3:26 // An example using L'Hospital's rule
5:13 // When you can and can't use L'Hospital's rule
7:35 // What to do when L'Hospital's rule fails
7:51 // Two examples that avoid L'Hospital's rule
10:11 // Another example using L'Hospital's rule
L'Hospital's rule is so named because i was discovered (maybe) by the mathematician L'Hospital in the 1600's. While L'Hospital is credited with the rule, he worked closely with Johann Bernoulli, who claimed after L'Hospital's death that L'Hospital had paid him off in order to take credit for the rule.
But no matter who discovered it, L'Hospital's rule is a really helpful tool that you can use to solve limit problems. You want to use L'Hospital's rule when you evaluate a limit and the result is an indeterminate form, like 0/0 or infinity/infinity. For a limit that gives an indeterminate form, L'Hospital's rule says that you can replace the numerator and denominator of the original function with their respective derivatives. In other words, replace the numerator with its derivative, and replace the denominator with its derivative. Then use substitution to try to evaluate the limit again. Oftentimes, L'Hospital's rule will have simplified the function to the point where substitution no longer gives an indeterminate form, but instead a real-number answer.
And L'Hospital's rule can actually be applied multiple times. So even if you use it once, and then try substitution and you still get an indeterminate form, you can just try applying L'Hospital's rule again and again until you eventually do get a real-number answer. The only drawback is that there are a few instances in which L'Hospital's rule can't be used.