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5.5 notes on L’Hopital’s Rule
Consider lim
x1
ln( x)
ex  e
∞
This is called an indeterminate form. (Other forms: ∞ , 0 ∙ ∞, ∞ − ∞, 00 , 1∞ , ∞0 )
Today, we are going to learn a shortcut for calculating limits with an indeterminable form. We did not
learn it when we did limits because at that time you did not know how to take a derivative. Now that
you can take a derivative, you can use something called L’Hopital’s Rule, which states:
Suppose that f(a) = g(a) = 0, that f’(a) and g’(a) exist, and that g’(a)  0. Then
lim
x a
f ( x) f '(a)

.
g ( x) g '(a)
WARNING: Do not use the quotient rule!
Find
ex
lim
x  2 x
Step 1: Check to see if the limit is indeterminate.
Step 2: Take the derivative of the top and the bottom separately. Leave the top on top and the
bottom on the bottom!
Examples:
ln(x)
x x 2
1.
lim
2.
 x  1
lim
x1
x 1
3.
2x
lim
x 1  3x
4.
x2
lim x
x0 2  1
2
5.
lim x ln( x)
x 0 
*Need to rewrite as a fraction first!
6.
1 1 
lim   2 
x 0  x
x 
*Need to combine first!
7.
lim
8.
lim
x 2
x 0
x3  8
x2  4
1  x 1
x
The next example leads us to introduce a “stronger” form of L’Hopital’s rule. It says that L’Hopital’s Rule can
continued to be applied if you get an indeterminate form. Stop differentiating as soon as you get something else.
Remember that L’Hopital’s rule does not apply when either the numerator or denominator has a finite nonzero
limit. If you reach a point where one of the derivatives is zero and the other is not, then the limit in question is
either zero or infinity.
9.
1 
1
lim   x 
x 0 x
e 1 

x

 1  x 1 2 
10. lim 

x0
x2




3x 2  2 x  7
11. lim 3
x  x  5 x  4
12. lim
x
3x3  2 x  7
x2  5x  4
HOMEWORK
3
x 1
x 1
1.
lim
2.
lim
3.
x 
 8
lim  2


x 2  x  4
x2
4.
lim
5.
lim
6.
lim
x1
x2
x3
x2  4x  4
x3  12 x  16
x4
x2
x2
x e5 x
e2 x  1
x0
ex