6.6. L’Hospital’s Rule http://www.ck12.org
6.6 L’Hospital’s Rule
Learning Objectives
A student will be able to:
- Learn how to find the limit of indeterminate form( 0 / 0 )by L’Hospital’s rule.
If the two functionsf(x)andg(x)are both equal to zero atx=a,then the limit
xlim→agf((xx))cannot be found by directly substitutingx=a.The reason is because when we substitutex=a,the substitution will
produce( 0 / 0 ),known as anindeterminate form, which is a meaningless expression. To work around this problem,
we use L’Hospital’s rule, which enables us to evaluate limits of indeterminate forms.
L’Hospital’s Rule
If limx→af(x) =limx→ag(x) =0, andf′(a)andg′(a)exist, whereg′(a) 6 =0, then
xlim→agf((xx))=xlim→af′(x)
g′(x).The essence of L’Hospital’s rule is to be able to replace one limit problem with a simpler one. In each of the examples
below, we will employ the following three-step process:
- Check that limx→agf((xx))is an indeterminate form 0/ 0 .To do so, directly substitutex=aintof(x)andg(x).If
you getf(a) =g(a) = 0 ,then you can use L’Hospital’s rule. Otherwise, it cannot be used. - Differentiatef(x)andg(x)separately.
- Find limx→agf′′((xx)).If the limit is finite, then it is equal to the original limit limx→agf((xx)).
Example 1:
Find limx→ 0
√ 2 +x−√ 2
x.
Solution:
Whenx=0 is substituted, you will get 0/ 0.
Therefore L’Hospital’s rule applies: