CK-12-Calculus

6.6. L’Hospital’s Rule http://www.ck12.org

A broader application of L’Hospital’s rule is whenx=ais substituted into the derivatives of the numerator and the
denominator but both still equal zero. In this case, a second differentiation is necessary.
Example 6:
Evaluate limx→ 0 ex−xcos 2 x.
Solution:

limx→ 0 e

x−cosx

x^2 =xlim→ 0

ex+sinx

2 x.

As you can see, if we apply the limit at this stage the limit is still indeterminate. So we apply L’Hospital’s rule again:

=limx→ 0 e

x−cosx

2

=^1 − 21 =^02 = 0.

Review Questions

Find the limits.

1. 
   

   limθ→ 0 tanθθ

   
   


2. 
   

   limx→ (^1) tanlnxπx

   
   


3. 
   

   limx→ 0 e^10 x−xe^6 x

   
   


4. 
   

   limx→πsinx−πx

   
   


5. 
   

   limx→ 01 xe−exx

   
   


6. 
   

   Ifkis a nonzero constant andx> 0.
   a. Show that∫ 1 xt 11 −kdt=xk−k^1.
   b. Use L’Hospital’s rule to find limk→ 0 xk−k^1.
   7.Cauchy’s Mean Value Theoremstates that if the functionsfandgare continuous on the interval(a,b)and
   g′ 6 = 0 ,then there exists a numbercsuch that
   f′(c)
   g′(c)=

   
   

f(b)−f(a)

g(b)−g(a).

Find all possible values ofcin the interval(a,b)that satisfy this property for

f(x) =cosx

g(x) =sinx

on the interval

[a,b] =

[

0 ,π 2

]

.

Texas Instruments Resources

In the CK-12 Texas Instruments Calculus FlexBook® resource, there are graphing calculator activities designed
to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9731.