1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
65
1.7 Limits, Continuity, and Differentiability
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What does it mean graphically to say that f has limit L as x → a? How is this
connected to having a left-hand limit at x = a and having a right-hand limit at
x = a?
• What does it mean to say that a function f is continuous at x = a? What role do
limits play in determining whether or not a function is continuous at a point?
• What does it mean graphically to say that a function f is differentiable at x = a?
How is this connected to the function being locally linear?
• How are the characteristics of a function having a limit, being continuous, and
being differentiable at a given point related to one another?
Introduction
In Section 1.2, we learned about how the concept of limits can be used to study the trend
of a function near a fixed input value. As we study such trends, we are fundamentally
interested in knowing how well-behaved the function is at the given point, say x = a.
In this present section, we aim to expand our perspective and develop language and
understanding to quantify how the function acts and how its value changes near a
particular point. Beyond thinking about whether or not the function has a limit L at
x = a, we will also consider the value of the function f (a) and how this value is related to
lim x→a f (x), as well as whether or not the function has a derivative f ′ (a) at the point of
interest. Throughout, we will build on and formalize ideas that we have encountered in
several settings.
We begin to consider these issues through the following preview activity that asks you
to consider the graph of a function with a variety of interesting behaviors.
Preview Activity 1.7. A function f defined on −4 < x < 4 is given by the graph in
Figure 1.35. Use the graph to answer each of the following questions. Note: to the right
of x = 2, the graph of f is exhibiting infinite oscillatory behavior similar to the function
sin(
π
x ) that we encountered in the key example early in Section 1.2.
(a) For each of the values a = −3, −2, −1, 0, 1, 2, 3, determine whether or not lim
x→a
f (x)
exists. If the function has a limit L at a given point, state the value of the limit
using the notation lim
x→a
f (x) = L. If the function does not have a limit at a given
point, write a sentence to explain why.
65
1.7 Limits, Continuity, and Differentiability
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What does it mean graphically to say that f has limit L as x → a? How is this
connected to having a left-hand limit at x = a and having a right-hand limit at
x = a?
• What does it mean to say that a function f is continuous at x = a? What role do
limits play in determining whether or not a function is continuous at a point?
• What does it mean graphically to say that a function f is differentiable at x = a?
How is this connected to the function being locally linear?
• How are the characteristics of a function having a limit, being continuous, and
being differentiable at a given point related to one another?
Introduction
In Section 1.2, we learned about how the concept of limits can be used to study the trend
of a function near a fixed input value. As we study such trends, we are fundamentally
interested in knowing how well-behaved the function is at the given point, say x = a.
In this present section, we aim to expand our perspective and develop language and
understanding to quantify how the function acts and how its value changes near a
particular point. Beyond thinking about whether or not the function has a limit L at
x = a, we will also consider the value of the function f (a) and how this value is related to
lim x→a f (x), as well as whether or not the function has a derivative f ′ (a) at the point of
interest. Throughout, we will build on and formalize ideas that we have encountered in
several settings.
We begin to consider these issues through the following preview activity that asks you
to consider the graph of a function with a variety of interesting behaviors.
Preview Activity 1.7. A function f defined on −4 < x < 4 is given by the graph in
Figure 1.35. Use the graph to answer each of the following questions. Note: to the right
of x = 2, the graph of f is exhibiting infinite oscillatory behavior similar to the function
sin(
π
x ) that we encountered in the key example early in Section 1.2.
(a) For each of the values a = −3, −2, −1, 0, 1, 2, 3, determine whether or not lim
x→a
f (x)
exists. If the function has a limit L at a given point, state the value of the limit
using the notation lim
x→a
f (x) = L. If the function does not have a limit at a given
point, write a sentence to explain why.