3.5 Limits Involving Infinity North Dakota Sunset

What you’ll learn about Finite Limits as x → ±∞ Sandwich Theorem Revisited Infinite Limits as x → a End Behavior Models Seeing Limits as x → ±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.

Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

Horizontal Asymptote

[-6,6] by [-5,5] Example Horizontal Asymptote

As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or

This number becomes insignificant as. There is a horizontal asymptote at 1.

Find:

Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative. vertical asymptote at x =0.

The denominator is positive in both cases, so the limit is the same.

Often you can just “think through” limits. 

Quick Quiz Slide 2- 12

Quick Quiz Slide 2- 13

Quick Quiz Slide 2- 14

Quick Quiz Slide 2- 15

Quick Quiz Slide 2- 16

Practice TEXT p. 88, #9 – 52; p.89, # 59, 62, 63 p. 205, #3 – 8, 15 – 20, 21 – 33 odds, 86 – 88. 