L'Hôpital's rule is a rule in calculus that helps find limits using derivatives. The name of the rule comes from the mathematician Guillaume de l'Hôpital[1]. L'Hôpital's rule states that for any function f whose limit as f approaches some point c is an indeterminate form the value of the limit of the function is equal to the derivative of the bottom and the top.[2][3] It is shown as follows:
Given the limit first plug in 5 to try to find the limit. Plugging in 5 will give an indeterminate form. Then L'Hôpital's rule is applied to give . Then simply plugging in 5 again yields the answer of 10.
The next example involves the sine function
plugging in 0 gives an indeterminate form
L'Hôpital's rule is applied and the derivatives of the top and bottom are taken
Plugging in 0 to the new expression gives 1, the answer to the limit
This example involves the natural log function
Plugging in 1 to the expression gives an indeterminate form
L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.
expression is simplified
Plugging back in 1 gives the answer to the limit as 1/2
The next example includes both sine and natural log functions
Plugging in 1 to the expression gives an indeterminate form
L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.
expression is simplified
Plugging back in 1 gives the answer to the limit as 2
The final example involves the tangent function
Plugging in 0 to the expression gives an indeterminate form
L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.
Plugging back in 0 gives the answer to the limit as 1