1
Lesson Plan #94
Name:
Class: PreCalculus
Date: Friday May 17th, 2013
Topic: Limits at Infinity
Aim: How do we determine the limits at infinity?
Objectives:
1) Students will be able to evaluate limits at infinity.
2) Students will be able to find the horizontal asymptotes of a function.
HW# 94:
Note:
Properties of Limits:
If b and c are real numbers, n is an integer, and the functions f and g have limits as x  c, then the following properties are true.
1. Scalar multiple:
lim [b  f ( x)]  b[lim f ( x)]
2. Sum or difference:
lim [ f ( x)  g ( x)]  lim f ( x)  lim g ( x)
x c
x c
x c
x c
x c
lim [ f ( x) g ( x)]  [lim f ( x)][ lim g ( x)]
x c
x c
x c
3. Product:
4. Quotient:
lim
x c
f ( x)
f ( x) lim
 x c
g ( x) lim g ( x)
x c
5. Power
lim [ f ( x)]n  [lim f ( x)]n
x c
Do Now:
1)
x c
where
lim g ( x)  0
x c
2
2) Sketch the graph of f ( x ) 
1
x
3) State the vertical asymptote(s) of f ( x ) 
1
x
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
1) In the function f ( x ) 
towards
.
2) In the function f ( x ) 
towards
.
1
, what value is the function approaching as x gets bigger and bigger, in other words, as x goes
x
1
, what value is the function approaching as x gets smaller and smaller, in other words, as x goes
x
3) Find the limits
A) lim
x
1
x
B) lim
x  
The limits in 3) are known as limits
In the graph of f ( x ) 
1
x
at infinity.
1
, where do we have a horizontal asymptote?
x
Notice that this is the same value that we got for the limits at infinity of this function.
Definition of a horizontal asymptote: If lim f ( x)  L or lim f ( x)  L , then the line y  L is called the
x  
horizontal asymptote of the graph of f
x 
3
c
0.
x  x r
Theorem: Limits at Infinity: If r is a positive integer and rational and c is any real number, then lim
Furthermore, if x r is defined when x > 0, then lim
x  
c
0
xr
Example #1:
Evaluate lim (5 
x 
2
)
x2
Example #2:
2x 1
{Hint: direct substitution yields an indeterminate form. Instead divide numerator and denominator by the
x  x  1
highest power of x in the denominator and then find limit as x 
Evaluate lim
Exercises:
I. Find the indicated limits
2x 1
x  3 x  2
1) lim
x
x  x  1
2) lim
3)
2
5x 2
x   x  3
lim
4) lim 2 x 
x 
5)
1
x2
5x 3  1
x  10 x 3  3 x 2  7
lim
II. Find the horizontal asymptote(s) of the function
1) f ( x) 
2 x
1 x
2) f ( x)  2 
3
x2
4
Sample Test Questions:
1)
4  x2
x 4 x 2  x  2
lim
A) -2
2)
B) 
1
4
C) 1
D) 2
E) None of the other choices
C) 0
D) 3
E) 
5 x3  27
x 20 x 2  10 x  9
lim
A)  
3) The graph of
B) -1
y
x2  9
has
3x  9
A) one vertical asymptote at
C) a removable discontinuity at
1
3
D) an infinite discontinuity at x  3
B) a horizontal asymptote at y 
x3
x3
E) none of the other choices
2x2  4
4) Which statement is true about the curve y 
?
2  7 x  4x2
1
is a vertical asymptote
4
1
C) The line y   is a horizontal asymptote
4
A) The line x  
B) The line x = 1 is a vertical asymptote
D) The graph has no vertical or horizontal asymptote
E) The line y = 2 is a horizontal asymptote
5)
2x2  1
is
x (2  x)( 2  x)
lim
A) -4
B) -2
C) 1
6) The graph of f ( x) 
D) 2
E) none of the other choices
4
has
x 1
2
A) One vertical asymptote at x = 1
B) The y-axis as a vertical asymptote
C) The x-axis as horizontal asymptote and x=  1 as vertical asymptotes
D) Two vertical asymptotes, at x  1 , but no horizontal asymptote
E) No asymptote
7) Let f (x ) 
x2  9
, x3
x 3
x 3
6,
Which of the following is true?
I. lim f ( x ) does not exist
x 3
III. The line
A) I only
B) II only
C) III only
II. f is continuous at
x  3 is a vertical asymptote
D) I and II only
E) I and III only
x 3