Download 11.4 - Limits at Infinity and Limits of Sequences

Ch. 11 – Limits and an
Introduction to Calculus
11.4 – Limits at Infinity and Limits
of Sequences
Rational Function Review
• Recall that rational functions (fractions with polynomials on
top and bottom) often have horizontal asymptotes
• Ex: Find the horizontal asymptotes of the following functions.
–
3x
f ( x)  2
x 1
• The degree of the top is smaller than the degree of the bottom
• Horizontal asymptote: y = 0
–
3x 2  2 x  6
f ( x) 
2x2  x
• The degree of the top equals the degree of the bottom
• To find horizontal asymptote, divide leading coefficients: y = 3/2
–
x2  4 x  1
f ( x) 
x 1
• The degree of the top is bigger than the degree of the bottom
• Divide top by bottom to get slant asymptote
2 x  3
• Ex: Find lim
x  3 x 2  1
.
– Use what you know about horizontal asymptotes…think of the graph!
– Since f(x) has a horizontal asymptote at y=0, f(x) will approach zero as x
approaches infinity
– Answer = 0
• Ex: Find
x2  3 .
lim
x  3 x  1
– Use what you know about horizontal asymptotes…think of the graph!
– Since f(x) has a slant asymptote at y=0, f(x) will approach infinity as x
approaches infinity
– Answer = ∞
• Ex: Find the limit of the sequence
2n  1
an 
n4
– Sequences go on forever, so we want to find
lim an
.
.
n 
– The degrees are equal on the top and bottom, so as n approaches
infinity, an approaches 2
8  n(n  1)(2n  1) 
• Ex: Find the limit of the sequence an  3 
.

n 
6

– Once again, find the limit as n approaches infinity…
– The degree of the numerator matches the degree of the denominator, so
find the ratio of the leading coefficients!
– When multiplied out, the leading coefficients are 16/6 = 8/3