Chapter 3: Limits and Continuity
Section 3.1: Limits
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 1. 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
±0.0001
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
Example: Find the following infinite limits.
(a) lim−
1
x−2
(b) lim+
1
x−2
x→2
x→2
1
x→2 x − 2
(c) lim
(d) lim
x→1
2−x
(x − 1)2
4
Theorem: (Limit Laws)
Suppose that c is a constant and the limits lim f (x) and lim g(x) exist. Then
x→a
x→a
1. lim [f (x) ± g(x)] = lim f (x) ± lim g(x)
x→a
x→a
x→a
2. lim [cf (x)] = c lim f (x)
x→a
x→a
3. lim [f (x)g(x)] = lim f (x) lim g(x)
x→a
x→a
x→a
lim f (x)
f (x)
= x→a
if lim g(x) 6= 0.
x→a g(x)
lim g(x) x→a
4. lim
x→a
h
in
5. lim [f (x)]n = lim f (x)
x→a
6. lim
x→a
x→a
p
n
f (x) =
q
n
lim f (x). If n is even, then we must have lim f (x) ≥ 0.
x→a
x→a
Example: Suppose that lim f (x) = 4 and lim g(x) = −2. Find
x→1
x→1
(a) lim [f (x) − 2g(x)]
x→1
(b) lim [g(x)]3
x→1
p
3 f (x)
(c) lim
x→1
g(x)
5
Example: Find the limit if it exists.
(a) lim (2x2 − 3x + 4)
x→5
x3 + 2x2 − 1
x→−2
5 − 3x
(b) lim
(c) lim−
√
9 − x2
x→3
(d) lim+
√
9 − x2
x→3
6
x2 + x − 12
x→3
x−3
(e) lim
x4 − 16
x→2 x − 2
(f) lim
√
(g) lim
x→0
1 + 3x − 1
x
1 1
−
(h) lim x 2
x→2 x − 2
7