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We have:
lim
x→a
f
(x)
g
(x)
=
f
(a)
g
(a)
=
lim
x→a
f(x) − f(a)
x − a
lim
x→a
g(x) − g(a)
x − a
= lim
x→a
f(x) − f(a)
g(x) − g(a)
= lim
x→a
f(x)
g(x)
.
It is more difficult to prove the general version of l’Hospital’s
Rule.
Example 22
Find lim
x→1
ln x
x − 1
.
Since lim
x→1
ln x =ln1=0and lim
x→1
(x − 1) = 0, we can apply
l’Hospital’s Rule:
lim
x→1
ln x
x − 1
= lim
x→1
1/x
1
= lim
x→1
1
x
=1.
Example 23
Find lim
x→0
sin 5x
3x
.
lim
x→0
sin 5x
3x
= lim
x→0
(sin 3x)
(3x)
= lim
x→0
5cos5x
3
=
5
3
.
E x a m p l e 24 (l’Hospital’s Rule applied more than once)
Find lim
x→∞
e
x
x
2
.
lim
x→∞
e
x
x
2
= lim
x→∞
e
x
2x
= lim
x→∞
e
x
2
= ∞.
34
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