1.2. THE NOTION OF LIMIT
19
approach a particular fixed value.
• When we write lim
x→a
f (x) = L, we read this as saying “the limit of f as x approaches a
is L,” and this means that we can make the value of f (x) as close to L as we want by
taking x sufficiently close (but not equal) to a.
• If we desire to know lim
x→a
f (x) for a given value of a and a known function f , we can
estimate this value from the graph of f or by generating a table of function values that
result from a sequence of x-values that are closer and closer to a. If we want the exact
value of the limit, we need to work with the function algebraically and see if we can use
familiar properties of known, basic functions to understand how different parts of the
formula for f change as x → a.
• The instantaneous velocity of a moving object at a fixed time is found by taking the
limit of average velocities of the object over shorter and shorter time intervals that all
contain the time of interest.
Exercises
1. Consider the function whose formula is f (x) =
16 − x 4
x 2 − 4
.
(a) What is the domain of f ?
(b) Use a sequence of values of x near a = 2 to estimate the value of lim
x→2
f (x), if
you think the limit exists. If you think the limit doesn’t exist, explain why.
(c) Evaluate lim x→2 f (x) exactly, if the limit exists, or explain how your work
shows the limit fails to exist. Here you should use algebra to factor and simplify
the numerator and denominator of f (x) as you work to evaluate the limit.
Discuss how your findings compare to your results in (b).
(d) True or false: f (2) = −8. Why?
(e) True or false:
16−x 4
x 2 −4
= −4 − x 2 . Why? How is this equality connected to your
work above with the function f ?
(f) Based on all of your work above, construct an accurate, labeled graph of
y = f (x) on the interval [1, 3], and write a sentence that explains what you
now know about lim
x→2
16 − x 4
x 2 − 4
.
2. Let g(x) = −
|x + 3|
x + 3
.
(a) What is the domain of g?
(b) Use a sequence of values near a = −3 to estimate the value of lim x→−3 g(x), if
you think the limit exists. If you think the limit doesn’t exist, explain why.
19
approach a particular fixed value.
• When we write lim
x→a
f (x) = L, we read this as saying “the limit of f as x approaches a
is L,” and this means that we can make the value of f (x) as close to L as we want by
taking x sufficiently close (but not equal) to a.
• If we desire to know lim
x→a
f (x) for a given value of a and a known function f , we can
estimate this value from the graph of f or by generating a table of function values that
result from a sequence of x-values that are closer and closer to a. If we want the exact
value of the limit, we need to work with the function algebraically and see if we can use
familiar properties of known, basic functions to understand how different parts of the
formula for f change as x → a.
• The instantaneous velocity of a moving object at a fixed time is found by taking the
limit of average velocities of the object over shorter and shorter time intervals that all
contain the time of interest.
Exercises
1. Consider the function whose formula is f (x) =
16 − x 4
x 2 − 4
.
(a) What is the domain of f ?
(b) Use a sequence of values of x near a = 2 to estimate the value of lim
x→2
f (x), if
you think the limit exists. If you think the limit doesn’t exist, explain why.
(c) Evaluate lim x→2 f (x) exactly, if the limit exists, or explain how your work
shows the limit fails to exist. Here you should use algebra to factor and simplify
the numerator and denominator of f (x) as you work to evaluate the limit.
Discuss how your findings compare to your results in (b).
(d) True or false: f (2) = −8. Why?
(e) True or false:
16−x 4
x 2 −4
= −4 − x 2 . Why? How is this equality connected to your
work above with the function f ?
(f) Based on all of your work above, construct an accurate, labeled graph of
y = f (x) on the interval [1, 3], and write a sentence that explains what you
now know about lim
x→2
16 − x 4
x 2 − 4
.
2. Let g(x) = −
|x + 3|
x + 3
.
(a) What is the domain of g?
(b) Use a sequence of values near a = −3 to estimate the value of lim x→−3 g(x), if
you think the limit exists. If you think the limit doesn’t exist, explain why.