Transcript Limits at Infinity

Limits Involving Infinity
Calculus Concepts
Approaching Positive Infinity

The function f has a limit L as x
increases without bound, written
lim f ( x)  L
x 
if the outputs f (x) can be made as
close to L as we like by taking x
sufficiently large.
Section 2.7
Approaching Negative Infinity

The function f has a limit L as x
decreases without bound, written
lim f ( x)  L
x 
if the outputs f (x) can be made as close
to L as we like by taking x sufficiently
small.
Section 2.7
Asymptotes

If lim f ( x)  b or
x  
lim f ( x)  b
x  
then y = b is called a horizontal asymptote
of the graph of f .
Section 2.7
Horizontal Asymptotes

If the degree of the numerator is smaller than
the degree of the denominator, then the
horizontal asymptote is y=0.
8x
f ( x)  2
x 1
Section 2.7

If the degree of the numerator is equal to the
degree of the denominator, then the horizontal
asymptote is quotient of the leading
coefficients.
2x
f ( x) 
x 1
Section 2.7

If the degree of the numerator is larger than the
degree of the denominator, there is a slant
asymptote. Look at the end behavior of the
asymptote.
x2
f ( x) 
x 1
Section 2.7
Examples
5x2  8x
lim 2
x  x  1
Section 2.7
Examples
x  6x
lim 3
x  3 x  x 2  1
4
Section 2.7
Examples
2 x3
lim 2
x  4 x  x  6
Section 2.7
Examples
6 x  4 x2
lim 2
x   5 x  x  6
Section 2.7
Examples
8 x3  4 x 2  3x  6
lim
3
x  
7x  2
Section 2.7
Section 2.7