AP CALCULUS AB Chapter 2: Limits and Continuity Section 2.2: Limits Involving Infinity

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

Section 2.2 – Limits Involving Infinity  To find Horizontal Asymptotes: Divide numerator and denominator by the highest power of x.  Note:

Example Sandwich Theorem Revisited

Properties of Limits as x→±∞

Product Rule: Constant Multiple Rule:

Properties of Limits as x→±∞

Infinite Limits as x→a

Vertical Asymptote

Example Vertical Asymptote [-6,6] by [-6,6]

Section 2.2 – Limits Involving Infinity  To find vertical asymptotes: 1. Cancel any common factors in the numerator and the denominator 2. Set the denominator equal to 0 and solve for x. The vertical asymptote is x=-1. (from denominator) There is a hole at x=2. (from the cancelled factor) The x-intercept is at x=-2. (from numerator)

End Behavior Models

Example End Behavior Models

End Behavior Models

Example “Seeing” Limits as x→±∞

Section 2.2 – Limits Involving Infinity  Definition of Infinite Limits: A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite limit.

Section 2.2 – Limits Involving Infinity c

c

 Properties of Infinite Limits If 1. Sum or difference: 2. Product: 3. Quotient: