Section 2.2: The Limit of a Function
Consider the function f defined by
f (x) =
sin x
.
x
This function is undefined at x = 0. However, we can take values of x as close to 0 as we
would like, provided that we never reach 0.
Let us evaluate f (x) for values of x close to 0.
x
±1
±0.5
±0.1
±0.01
±0.001
f (x)
0.84147098
0.95885108
0.99833417
0.99998333
0.99999983
As the values of x approach 0, the values of f (x) approach 1. In this case, we say that the
limit of f (x) as x approaches 0 is equal to 1. This is denoted by
sin x
= 1.
x→0 x
lim
The limit can also be found by examining the graph of f (x).
Definition: If the values of f (x) can be made arbitrarily close to L by taking x sufficiently
close (but not equal) to a, then we say that the limit of f (x), as x approaches a, equals L.
This is denoted by
lim f (x) = L.
x→a
1
Example: Consider the Heaviside function H defined by
0, x < 0
H(x) =
1, x ≥ 0.
Find lim H(x).
x→0
The graph of H(x) is given below.
As x approaches 0 from the left, H(x) approaches 0. Similarly, as x approaches 0 from
the right, H(x) approaches 1. In this case, we say that the left-hand limit of H(x) as x
approaches 0 equals 0. This is denoted by
lim H(x) = 0.
x→0−
Similarly, the right-hand limit of H(x) as x approaches 0 equals 0. This is denoted by
lim H(x) = 1.
x→0+
Since there is no single number L that H(x) approaches as x approaches 0 from both directions, the limit does not exist.
Definition: The left-hand limit of f (x), as x approaches a, is equal to L if we can make
the values of f (x) arbitrarily close to L by taking x sufficiently close to (and less than) a.
This is denoted by
lim− f (x) = L.
x→a
Similarly, the right-hand limit of f (x), as x approaches a, is equal to L if we can make
the values of f (x) arbitrarily close to L by taking x sufficiently close to (and greater than)
a. This is denoted by
lim+ f (x) = L.
x→a
2
Theorem: (One-Sided Limit Theorem)
The limit of f (x), as x approaches a, is equal to L if and only if
lim f (x) = lim+ f (x) = L.
x→a−
x→a
Example: The graph of a function f is given below. Find the following values, if they exist.
lim f (x) =
x→−2−
lim f (x) =
x→3−
lim f (x) =
lim f (x) =
x→−2
x→−2+
lim f (x) =
lim f (x) =
x→3
x→3+
1
.
x→0 x2
Example: Find lim
Using a table of values
x
±1
±0.1
±0.01
±0.001
±0.001
f (x)
1
102
104
106
108
In this case, the values of f (x) are increasing without bound. Therefore,
lim f (x) = ∞.
x→0
Since ∞ is not a number, the limit does not exist.
3
f (−2) =
f (3) =
Definition: If the values of f (x) can be made arbitrarily large by taking x sufficiently close
(but not equal) to a, then
lim f (x) = ∞.
x→a
Similarly, if the values of f (x) can be made arbitrarily large negative by taking x sufficiently
close (but not equal) to a, then
lim f (x) = −∞.
x→a
Definition: The line x = a is called a vertical asymptote of the curve y = f (x) if
lim f (x) = ±∞
x→a−
or
lim f (x) = ±∞,
x→a+
Note: The graph of a function y = f (x) has a vertical asymptote when the simplified form
of the function involves division by zero.
Example: Find the following infinite limits.
(a) lim−
1
x−2
(b) lim+
1
x−2
x→2
x→2
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(c) lim
1
x−2
(d) lim
2−x
(x − 1)2
x→2
x→1
Example: Find the vertical asymptotes of
f (x) =
x2
x−2
.
− 5x + 6
Describe the behavior of f (x) near each asymptote.
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