2.8. USING DERIVATIVES TO EVALUATE LIMITS
149
2.8 Using Derivatives to Evaluate Limits
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can derivatives be used to help us evaluate indeterminate limits of the form
0
0 ?
• What does it mean to say that lim x→∞ f (x) = L and lim x→a f (x) = ∞?
• How can derivatives assist us in evaluating indeterminate limits of the form
∞
∞ ?
Introduction
Because differential calculus is based on the definition of the derivative, and the definition
of the derivative involves a limit, there is a sense in which all of calculus rests on limits.
In addition, the limit involved in the limit definition of the derivative is one that always
generates an indeterminate form of
0
0 . If f is a differentiable function for which f ′ (x)
exists, then when we consider
f
′ (x) = lim
h→0
f (x + h) − f (x)
h
,
it follows that not only does h → 0 in the denominator, but also ( f (x + h) − f (x)) → 0 in
the numerator, since f is continuous. Thus, the fundamental form of the limit involved
in the definition of f ′ (x) is
0
0 . Remember, saying a limit has an indeterminate form only
means that we don’t yet know its value and have more work to do: indeed, limits of the
form
0
0 can take on any value, as is evidenced by evaluating f ′ (x) for varying values of x
for a function such as f ′ (x) = x 2 .
Of course, we have learned many different techniques for evaluating the limits that
result from the derivative definition, and including a large number of shortcut rules that
enable us to evaluate these limits quickly and easily. In this section, we turn the situation
upside-down: rather than using limits to evaluate derivatives, we explore how to use
derivatives to evaluate certain limits. This topic will combine several different ideas,
including limits, derivative shortcuts, local linearity, and the tangent line approximation.
Preview Activity 2.8. Let h be the function given by h(x) =
x 5 + x − 2
x 2 − 1
.
(a) What is the domain of h?
(b) Explain why lim
x→1
x 5 + x − 2
x 2 − 1
results in an indeterminate form.
149
2.8 Using Derivatives to Evaluate Limits
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can derivatives be used to help us evaluate indeterminate limits of the form
0
0 ?
• What does it mean to say that lim x→∞ f (x) = L and lim x→a f (x) = ∞?
• How can derivatives assist us in evaluating indeterminate limits of the form
∞
∞ ?
Introduction
Because differential calculus is based on the definition of the derivative, and the definition
of the derivative involves a limit, there is a sense in which all of calculus rests on limits.
In addition, the limit involved in the limit definition of the derivative is one that always
generates an indeterminate form of
0
0 . If f is a differentiable function for which f ′ (x)
exists, then when we consider
f
′ (x) = lim
h→0
f (x + h) − f (x)
h
,
it follows that not only does h → 0 in the denominator, but also ( f (x + h) − f (x)) → 0 in
the numerator, since f is continuous. Thus, the fundamental form of the limit involved
in the definition of f ′ (x) is
0
0 . Remember, saying a limit has an indeterminate form only
means that we don’t yet know its value and have more work to do: indeed, limits of the
form
0
0 can take on any value, as is evidenced by evaluating f ′ (x) for varying values of x
for a function such as f ′ (x) = x 2 .
Of course, we have learned many different techniques for evaluating the limits that
result from the derivative definition, and including a large number of shortcut rules that
enable us to evaluate these limits quickly and easily. In this section, we turn the situation
upside-down: rather than using limits to evaluate derivatives, we explore how to use
derivatives to evaluate certain limits. This topic will combine several different ideas,
including limits, derivative shortcuts, local linearity, and the tangent line approximation.
Preview Activity 2.8. Let h be the function given by h(x) =
x 5 + x − 2
x 2 − 1
.
(a) What is the domain of h?
(b) Explain why lim
x→1
x 5 + x − 2
x 2 − 1
results in an indeterminate form.