Transcript Concavity and the Second Derivative Test and Curve Sketching
Limits at Infinity
Section 3.5
1
Horizontal Asymptotes
The line y = L is a horizontal asymptote of the graph of f if
x
lim
L or x
f x
L
2
Example
3
x
1 2
x
5
x
lim 3
x
1 2
x
5
x
lim 3
x
1 2
x
5 Check out TABLE on calculator.
3
Limits at Infinity
If r is a positive rational number and c is any real
x
lim
x c r
0 Furthermore, if x r
c
0 .
x
lim
x r
is defined for x < 0, then 4
Examples
1) Find :
x
lim 3 2
x
2) Find :
x
lim 4
x
5 3
x
1 Strategy: Divide each term of the numerator and denominator by the highest power of x in the denominator. 5
Examples
5
x
3 3) Find : lim
x
x
2 1 4) Find :
x
lim 10
x
2 2
x
2 1 3 6
Examples
2
x
3 5) Find :
x
lim
x
7 2 7
Limits at Infinity
Guidelines for finding limits at infinity of rational functions
If degree of numerator is less than degree of denominator, the limit is zero.
If degree of numerator is equal to degree of denominator, the limit is the ratio of the leading coefficients.
If degree of numerator is greater than degree of denominator, the limit does not exist.
8
End Behavior
End behavior of the graph of functions: Example: Describe end behavior of
x
2 3
x
9
End Behavior
Example: Describe end behavior of 2
x
2
x
1
x
10
Homework
Section 3.5 page 205 #1, 3-8 all, 17, 21-25 odd, Full curve sketches for 55, 59, 65, 67 11