Math 151
Section 2.2
The Limit of a Function
Limits We can use limits of functions to describe how the values of a function behave. The
behavior of a function depends on the specific location and our approach to that location.
Definition of a Limit
If lim" f ( x) = L and lim+ f ( x) = L then lim f ( x ) is said to exist and lim f ( x) = L .
x!a
x!a
x!a
If lim" f ( x) # lim+ f ( x) then lim f ( x ) does not exist.
x!a
x!a
x!a
Example: Use the graph of f(x) to evaluate the following limits.
lim f ( x) =
x!"1"
lim f ( x) =
x!"1+
lim f ( x) =
x!"1
lim f ( x) =
x!1"
lim f ( x ) =
x!1
lim f ( x) =
x!5"
lim f ( x ) =
x!5
lim f ( x) =
x!"3"
lim f ( x) =
lim f ( x) =
x!1+
f(1) =
lim f ( x) =
x!5+
f(5) =
lim f ( x) =
x!"3
lim f ( x) =
lim f ( x ) =
x!4"
x!4+
lim f ( x) =
x!"#
x!"
lim f ( x) =
x!"3+
lim f ( x) =
x!4
x!a
Math 151
Example: Use the piecewise definition of g(x) to sketch a graph of g(x) and identify all values of x
for which the limit does not exist.
#
%
2 ! x if x < !1
%
%
%x
if !1" x <1
g ( x) = %
$
%
4
if x = 1
%
%
%
%
&4 ! x if x >1
Vertical Asymptote The line x = a is said to be a vertical asymptote of a function f(x) if
lim" f ( x) = ±# or lim+ f ( x) = ±" .
x!a
x!a
Example: For what values of x does the f(x) have a vertical asymptote in example on p. 1?
Example: Find the following limits.
1
=
x!5 x "5
A. lim
Math 151
x "1
=
x!0 x
( x + 2)
B. lim
2
C. lim csc x =
x!!
Example: Find all vertical asymptotes of f ( x) =
each vertical asymptote.
x +1
. Describe the behavior of f(x) near
x 2 ! 2x ! 3