1. Chapter 3: Applications of Differentiation L3.5 Limits at Infinity

2. Limits at Infinity End behavior of a function over an infinite interval left end behavior right end behavior If a function grows without bound or oscillates as x→ − ∞ or x→∞, then dne If the line y = L is a horizontal asymptote of the graph of f, then or The above implies that a function can have up to two horizontal asymptotes (HAs) –Rational functions have at most one HA –Functions that are not rational can have up to two HAs

3. Limits at Infinity: Rational Functions Rational Functions have up to one horizontal asymptote: –HA is y = 0 if deg(numerator) < deg(denominator) –HA is y = ‘ratio of leading coefficients’ if deg(num) = deg(denom) –Otherwise, no horizontal asymptote Examples: 1. a. b. 2. 3. 4.

4. Limits at Infinity: Non Rational Functions Functions that are not rational (include ) may approach different horizontal asymptotes at the left and right ends. Example: For x > 0,. So For x < 0,. So Recall that