L'Hopital's Rule
Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.
Other "Indeterminate Forms"
" " is called an indeterminate form because knowing that approaches and approaches is not enough to determine the limit of , even if it has a limit. The ratio of a "small" number divided by a "small" number can be almost anything as the three simple " " examples show:
Similarly, " " is an indeterminate form because knowing that and both grow arbitrarily large is not enough to determine the value limit of or if the limit exists:
Besides the indeterminate quotient forms " " and " " there are several other "indeterminate forms". In each case, the resulting limit depends not only on each function's limit but also on how quickly each function approaches its limit.
Product: If approaches , and grows arbitrarily large, the product has the indeterminant form " ".
Exponent: If and both approach , the function has the indeterminant form " ".
If approaches , and g grows arbitrarily large, the function has the indeterminant form " ".
If grows arbitrarily large, and approaches , the function has the indeterminant form " ".
Difference: If and both grow arbitrarily large, the function has the indeterminant form " ".
Unfortunately, l'Hô pital's Rule can only be used directly with an indeterminate quotient (" " or " '), but these other forms can be algebraically manipulated into quotients, and then l'Hô pital's Rule can be applied to the resulting quotient.
Example 5: Evaluate (" " form)
Solution: This limit involves an indeterminate product, and we need a quotient in order to apply l'Hô pital's Rule. We can rewrite the product as the quotient , and then so apply l'Hô pital's Rule
A product with the indeterminant form " " can be rewritten as a quotient, or , and then l'Hô pital's Rule can be used.
Solution: An indeterminate exponent can be converted to a product by recalling a property of exponential and
logarithm functions: for any positive number so .
and this last limit involves an indeterminate product which we converted to a quotient and evaluated to be in Example 5.
Our final answer is then :
An indeterminate form involving exponents, with the form " , " " , " or " , " can be converted to an indeterminate product by recognizing that and then determining the limit of . The final result is .
Example 7: Evaluate (" " form)
Solution: so we need " " an indeterminate product so rewrite it as a quotient
an indeterminate quotient so use l'Hô pital's Rule