Page 1 | © 2012 by Janice L. Epstein 2.2 The Limit of a Function Page 2 | © 2012 by Janice L. Epstein 2.2 The Limit of a Function The line x = a is called a vertical asymptote of the function f ( x)
if at least one of the following is true:
The Limit of a Function (Section 2.2)
The limit of f ( x) as x approaches a is L if we can make the values
of f ( x) arbitrarily close to L as x is close to a but not equal to a.
lim f ( x) = L
lim f ( x) = ¥ or -¥
x  a-
lim f ( x) = ¥ or -¥
x a +
x a
5
f ( x)
What can we say about the f ( x)
in the graph on the left?
EXAMPLE 1
Using the graph of f ( x) find the following limits, if they exit
lim f ( x ) is the left-hand limit
a) lim f ( x)
x  a-5
5
x
lim f ( x) is the right-hand limit
x a +
x-5
b) lim f ( x)
x-3
c) lim f ( x)
x-2
lim f ( x) = L if and only if lim- f ( x) = L and lim+ f ( x) = L
x a
x a
x a
d) lim f ( x)
x 0
Let f be a function defined on both sides of a, except possibly at a
itself. Then lim f ( x) = ¥ means that the values of f ( x) can be
e) lim f ( x)
x1
x a
made arbitrarily large by taking x sufficiently close to a.
Similarly, lim f ( x) = -¥ when the values of f ( x) can be made
x a
arbitrarily negatively large by taking x sufficiently close to a.
f) lim f ( x)
x 4
Page 3 | © 2012 by Janice L. Epstein 2.2 The Limit of a Function Page 4 | © 2012 by Janice L. Epstein g
EXAMPLE 2
2.2 The Limit of a Function EXAMPLE 3
Evaluate the function g ( x) at the given values and then guess the
value of lim+ g ( x) .
Sketch the graph of the function
x 2
ì2 - x
ï
ï
ï
ïx
g ( x) = ï
í
ï
4
ï
ï
ï
ï
î4 - x
if x < -1
if -1 £ x < 1
1- x 2
g ( x) = 2
x + 3 x -10
2
if x = 1
if x > 1
2
Use the graph to find the following limits, if they exist:
a) lim g ( x)
x-1
b) lim g ( x)
x1
x
g (3) =
g (2.001) =
g (2.1) =
g (2.0001) =
g (2.01) =
g (2.00001) =
EXAMPLE 4
Evaluate the function g ( x) at the given values and then guess the
value of lim- g ( x) .
x0
cos x -1
g ( x) =
sin x
g (-1) =
g (-0.1) =
g (-0.5) =
g (-0.01) =
g (-0.3) =
g (-0.001) =
Page 5 | © 2012 by Janice L. Epstein 2.2 The Limit of a Function Page 6 | © 2012 by Janice L. Epstein 2.2 The Limit of a Function Limits of Vector Functions: lim r (t ) = b
EXAMPLE 5
Find the infinite limits
a) lim
x0
t a
If r (t ) = f (t ), g (t ) , then lim r (t ) = lim f (t ),lim g (t )
x -1
2
x ( x + 2)
ta
ta
t a
EXAMPLE 7
b) lim
x5
t 2 - 4t + 4 t 2 - 4
, find lim r (t ) by finding
Given r (t ) =
,
t 2
t -2
2t - 4
6
5- x
r (1.2) =
r (2.8) =
EXAMPLE 6
r (1.5) =
r (2.5) =
x
Find the vertical asymptotes of y = 2
and sketch the
x - x-2
graph.
r (1.8) =
r (2.2) =
r (1.9) =
r (2.1) =
r (1.99) =
r (2.01) =
5
-5
5