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of type
∞
∞
. For certain functions, this type of limit can be evaluated
by dividing numerator and denominator by the highest power of x
that occurs. For instance,
lim
x→∞
x
2
− 1
2x
2
+1
= lim
x→∞
1 − 1/x
2
2+1/x
2
=
1 − 0
2+0
=
1
2
.
This method does not work for limits such as (4), but l’Hospital’s
Rule also applies to this type of indeterminate forms.
L’Hospital’s Rule. Suppose f and g are differentiable and g
(x) =
0 near a (except possibly at a). Suppose that lim
x→a
f(x)=0and
lim
x→a
g(x)=0or that lim
x→a
f(x)=±∞ and lim
x→a
g(x)=±∞.(Inother
words, we have an indeterminate form of type
0
0
or
∞
∞
.) Then
lim
x→a
f(x)
g(x)
= lim
x→a
f
(x)
g
(x)
,
if the limit on the right side exists (or is ∞ or −∞).
Remark 10
L’Hospital’s Rule says that the limit of a quotient of functions is
equal to the limit of the quotient of their derivatives. It is important
to verify the conditions regarding the limits of f and g before using
l’Hospital’s Rule.
Remark 11
L’Hospital’s Rule is also valid for one-sided limits and for limits
at infinity or negative infinity; that is, “x → a” can be replaced by
any of the following symbols: x → a
+
, x → a
−
, x →∞, x →−∞.
For the special case in which f(a)=g(a)=0,f
and g
are
continuous, and g
(a) =0, it is easy to prove the l’Hospital’s Rule.
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