L'Hospital's Rule

L'Hospital's Rule is useful when determining the behavior of a function having a limit of indeterminate form. This discussion will focus on two types of indeterminate forms.

The first type occurs when the limit of a function results in a limit of 0 0 and is said to be a limit of indeterminate form 0 0 .

INDETERMINATE FORM 0 0 :


lim x1 logx x1 = log1 11 = 0 0




The second type occurs when the limit of a function results in a limit of and is said to be a limit of indeterminate form .

INDETERMINATE FORM :


lim x logx x1 = log 1 =




Let f(x) and g(x) be two functions that are differentiable on an open interval with point s, except maybe at s, and the first derivative of g(x) is not zero. If the limit of f( x ) g( x ) as x approaches s is of indeterminate form 0 0 or then L'Hospital's Rule can be applied.

L'Hospital's Rule states that the limit of the quotient of the two functions is equal to the limit of the quotients of their first derivatives.

L'HOSPITAL'S RULE:


If f(x) and g(x) are differentiable on an open interval that contains s (except maybe at s) and the limit is of indeterminate form 0 0 or , that is:


lim xs f( x )=0 AND lim xs g( x )=0


OR


  lim x f( x )=      AND lim x g( x )=


Then the following rule applies:


lim xs f( x ) g( x ) = lim xs f ( x ) g ( x )




Let's look at some examples.

To work these examples requires the use of various derivative rules. If you are not familiar with a rule go to the associated topic for a review.


Example 1: Find lim x2 x 2 2x x 3 +x10 .

Step 1: Ensure that the limit is of indeterminate form 0 0 or .

Take the limit:


lim x2 x 2 2x x 3 +x10


lim x2 ( 2 ) 2 2( 2 ) ( 2 ) 3 +( 2 )10 = 0 0

Step 2: Apply L'Hospital's Rule


Because the limit is of indeterminate form 0 0 , L'Hospital's rule can be applied.


lim xs f( x ) g( x ) = lim xs f ( x ) g ( x )

lim x2 x 2 2x x 3 +x10


lim x2 2x2 3 x 2 +1 Take the derivative


lim x2 2( 2 )2 3 ( 2 ) 2 +1 = 2 13 Take the limit

Example 1: Find lim x lnx x 3 +1 .

Step 1: Ensure that the limit is of indeterminate form 0 0 or .

Take the limit:


lim x lnx x 3 +1


lim x lnx ( ) 3 +1 =

Step 2: Apply L'Hospital's Rule


Because the limit is of indeterminate form , L'Hospital's rule can be applied.


lim xs f( x ) g( x ) = lim xs f ( x ) g ( x )

lim x lnx x 3 +1


lim x 1/x 3 x 2 = 1 3 x 3    Take the derivative


lim x 1 3 x 3 = 1 3 ( ) 3 =0    Take the limit





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