Page 136 - Calculus Demystified

CHAPTER 5

Indeterminate

Forms

5.1 l’Hôpital’s Rule

5.1.1 INTRODUCTION
Consider the limit

f(x)
lim . (∗)
x→c g(x)

If lim x→c f(x) exists and lim x→c g(x) exists and is not zero then the limit
(∗) is straightforward to evaluate. However, as we saw in Theorem 2.3, when
lim x→c g(x) = 0 then the situation is more complicated (especially when
lim x→c f(x) = 0 as well).
For example, if f(x) = sin x and g(x) = x then the limit of the quotient as
2
x → 0 exists and equals 1. However if f(x) = x and g(x) = x then the limit of
the quotient as x → 0 does not exist.
In this section we learn a rule for evaluating indeterminate forms of the type (∗)
wheneitherlim x→c f(x) = lim x→c g(x) = 0orlim x→c f(x) = lim x→c g(x) =∞.
Such limits, or “forms,” are considered indeterminate because the limit of the quo-
tient might actually exist and be finite or might not exist; one cannot analyze such
a form by elementary means.

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