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denominator is 0. In fact, although the limit in (3) exists, its value
is not obvious because both numerator and denominator approach 0
and 0/0 is not defined.
In general, if we have a limit of the form lim
x→a
f(x)
g(x)
where both
f(x) → 0 and g(x) → 0 as x → a, then this limit may or may not
exist and is called an indeterminate form of type 0/0.Youcould
have met some limits before. For rational functions, we can cancel
common factors:
lim
x→1
x
2
− x
x − 1
= lim
x→1
x(x − 1)
(x −1)(x +1)
= lim
x→1
x
x +1
=
1
2
.
A geometric argument can be used to show lim
x→0
sin x
x
=1.
But these methods do not work for limits such as (3), so we now
introduce a systematic method, known as l’Hospital’s Rule, for the
evaluation of indeterminate forms.
Another situation in which a limit is not obvious occurs when
we look for a horizontal asymptote of f(x) and need to evaluate the
limit
lim
x→∞
ln x
x − 1
. (4)
It is not obvious how to evaluate this limit because both numerator
and denominator become large as x →∞. There is a struggle
between numerator and denominator. If the numerator wins, the
limit will be ∞; if the denominator wins, the answer will be 0. Or
there may be some compromise, in which case the answer may be
some finite positive number.
In general, if we have a limit of the form
lim
x→a
f(x)
g(x)
,
where both f(x) → +∞ (or −∞)andg(x) → +∞ (or −∞), then
the limit may or may not exist and is called an indeterminate form
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