Section 2.2a

Limits Involving Infinity We can say “the limit of f as x approaches infinity,” meaning the limit of f as x moves increasingly far to the right on the number line, or… Note: “Infinity” does not represent a real number……however, Saying “the limit of f as x approaches negative infinity” means the limit of f as x moves increasingly far to the left.

Limits Involving Infinity The graph of the reciprocal function: Our new limits: The line y = 0 is a horizontal asymptote of the graph of f…

Definition: Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either or Ex: Find any H.A. of the graph of  H.A.: y = 2

Definition: Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either or Ex: Find any H.A. of the graph of  H.A.: y = –1, y = 1 Investigate with both a graph and a table…

Sandwich Theorem Revisited Findfor First, what do the graph and table suggest??? Confirm Analytically:For x > 0, we have And by the Sandwich Theorem:

Limits Involving Infinity Properties of Our New Limits: Note: All of the properties for limits approaching real numbers also hold for limits approaching infinity!!! Including  Sum Rule, Difference Rule, Product Rule, Constant Multiple Rule, Quotient Rule, Power Rule

Limits Involving Infinity Find Rewrite:

Limits Involving Infinity Sometimes, a function outgrows all bounds (either positive or negative) as x approaches a finite number a  we write: or Think back to the reciprocal function: and The line x = 0 is a vertical asymptote of the graph of f…

Definition: Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either or

Limits Involving Infinity For the given function, (a) find the vertical asymptotes; (b) describe the behavior of the function to the left and right of each V.A. Now, check the graph! V.A.: