Example 3 Using Properties of Limits
5x + sin x
=
x →∞
x
a)
lim
b)
 3x 2 + 1
 =
lim
x →∞
x2 
c)
  1
lim cos   =
x →∞
 x 
B. Infinite Limits as x → a
Look back at the graph of y =
1
. Determine:
x
1
=
x →0+ x
a) lim
b) lim
x →0−
1
=
x
Definition: Vertical Asymptote
The line x = a is a vertical asymptote of the graph of a function y = f ( x ) if either
lim f ( x ) = ±∞ or lim f ( x ) = ±∞
x →a+
x →a−
Calculus Section 2.2 Page9
Example 4 Finding Vertical Asymptotes
Find the vertical asymptotes for each function (use a graph and limits to help):
1
a) f (x ) = 2
x
b) f (x ) =
1
x +1
c) f (x ) =
1
x −1
2
d) f (x ) = tan x
Epp short-cut for finding Vertical Asymptotes
Calculus Section 2.2 Page10
C. End Behaviour Models
Example 5 Modeling Functions for x Large
Let f (x ) = 3x 4 − 2x 3 + 3x 2 − 5x + 6 and g(x ) = 3x 4 . Show that while f and g are quite
different for numerically small values of x, they are virtually identical for x large.
Solve Graphically
Confirm Analytically
Definition: End Behaviour Model
f (x )
=1
x →∞ g( x )
f (x )
The function g is a left end behaviour model for f if and only if lim
=1
x →−∞ g( x )
The function g is a right end behaviour model for f if and only if lim
Example 6 Finding End Behaviour Models
Let f (x ) = x + e −x . Show that g(x ) = x right end behaviour model for f while h(x ) = e − x
is a left end behaviour model for f.
Solve Graphically
Confirm Analytically
Calculus Section 2.2 Page11
Example 7 Finding End Behaviour Models
Find an end behaviour model for
a) f (x ) =
2x 5 + x 4 − x 2 + 1
2
3x − 5x + 7
b) f (x ) =
2x 3 − x 2 + x − 1
3
2
5x + x + x − 5
Example 8 Using the End Behaviour Models to find limits as x → ±∞
Evaluate:
2x 5 + x 4 − x 2 + 1
2
x →∞
3x − 5x + 7
2x 5 + x 4 − x 2 + 1
2
x →−∞
3x − 5x + 7
2x 3 − x 2 + x − 1
3
2
x →∞ 5x + x + x − 5
2x 3 − x 2 + x − 1
3
2
x →−∞ 5x + x + x − 5
4x 2 − 3x + 5
3
x →∞ 2x + x − 1
4x 2 − 3x + 5
lim
3
x →−∞ 2x + x − 1
a) lim
b) lim
lim
lim
c) lim
Question: Which of the functions above have horizontal asymptotes? Why?
Epp Short-cut…..
Calculus Section 2.2 Page12