Infinite Limits
Lesson 3.5
Infinite Limits
Infinite Limits – a limit in which f (x) increases or decreases
without bound as x approaches c.
1
1
Evaluate the limits lim
lim
.
and 
x 3 x  3
x 3 x  3
As x  3 from
the left…
As x  3 from
the right…
x
2.9
2.99 2.999
3.001 3.01
3.1
f (x)
-10
-100 -1,000 undef. 1,000 100
10
f (x)  –∞
1
lim
 
x 3 x  3
3
f (x)  ∞
1
lim

x 3 x  3
Determining Infinite Limits from Graphs
Determine each function’s limit as x approaches 1 from the left and right.
1) f ( x) 
1
x 1
lim f (x) = -∞
-
x→1
lim f (x) = ∞
+
x→1
2) g ( x) 
1
( x  1) 2
lim g(x) = ∞
-
x→1
lim g(x) = ∞
+
x→1
3) h( x) 
1
x 1
lim h(x) = ∞
-
x→1
lim h(x) = -∞
+
x→1
4) k ( x) 
1
( x  1) 2
lim k(x) = -∞
x→1-
lim k(x) = -∞
x→1+
Vertical Asymptotes
asymptote – (review) a line that continually approaches the
graph of a function without ever meeting it at a finite distance
1
Ex: f ( x) 
has a vertical asymptote
x 3
of x = 3.
At x = 3, the denominator x– 3
equals 0, while the numerator
doesn’t. The v.a. exists @ 3.
Vertical Asymptote Theorem – a
rational function f (x)=g(x)/h(x) will
have a vertical asymptote at x = c if
h(c) = 0 while g(c) ≠ 0, as long as
g(x) and h(x) are continuous about c.
Determining Vertical Asymptotes
Determine all vertical asymptotes of the graph of each function.
x
1) f ( x)  2
x x
x
=
(x)(x -1)
2
x
2) g ( x)   1
x2 1
Denominator = 0
when x = 0 or 1…
x2 + 1
=
(x + 1)(x -1)
Denominator = 0
when x = -1 or 1…
When x = 0,
den. = 0 & num. = 0
 No v.a. @ x = 0
When x = 1,
den. = 0 & num. ≠ 0
 v.a. @ x = 1
When x = 1,
den. = 0 & num. ≠ 0
 v.a. @ x = 1
When x = -1,
den. = 0 & num. ≠ 0
 v.a. @ x = -1
Determining Infinite Limits
Properties of Infinite Limits – Let c and L be real numbers,
and let f and g be functions such that
lim f ( x)   & lim g ( x)  L . Then,
x c
x c
[ f ( x)  g ( x)]  
1) lim
x c
g ( x)
lim
0
3) xc f ( x)
[ f ( x) g ( x)]  , L  0
2) lim
x c
lim [ f ( x) g ( x)]  , L  0
x c
1
( x 2  1)
Find the limits of lim (1  2 ) and lim
x 0
x 1 1 /( x  1)
x
2
1
+ 1)
2
(
x
lim(1+
)
->
1+
∞
=
∞
lim
->
=
x->0
x2
->
x 1 1/(x -1)
-∞
0
Lesson Practice
1
1) Determine the limits of the function f ( x) 
3 x
as x approaches 3 from the left and right.
2) Find the vertical asymptotes of the function
f ( x) 
x3
x2  4x  3
Classwork
Pg. 243 (1-4, 9-12, 25-26, 29, 31, 33, 41, 43, 4546, 51-53)