2.2 Limits Involving Infinity, p. 70 AP Calculus AB/BC 2.2 Limits Involving Infinity, p. 70

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

Definition: Horizontal Asymptote The line y = b is a a horizontal asymptote of the graph of a function y = f(x) if either: For example: y = -3 is the horizontal asymptote.

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

Use graphs and tables to find: Example 1b Use graphs and tables to find: Identify all horizontal asymptotes. = 0 = -∞ y = 0 (the x-axis)

Use graphs and tables to find: Example 2 Use graphs and tables to find: Identify all horizontal asymptotes. = 0 = 0 y = 0 First, graph on your calculator. Next, use the table feature on your calculator to look at extremely large and small values for x.

Example 2 (cont.) Since then by the sandwich theorem:

Theorem 5: Properties of Limits Limits at infinity have properties similar to those of finite limits.

Example 3 Find: p Day 1

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

Finding Vertical Asymptotes The denominator of any function cannot equal zero. So, set the denominator equal to zero and solve. The result is/are a vertical asymptote.

Find the vertical asymptotes of the graph of f(x). Example 4 Find the vertical asymptotes of the graph of f(x). Describe the behavior of f(x) to the left and right of each vertical asymptote. Set x + 1 = 0. The vertical asymptote is the line x = -1.

Example 4 (cont.) To describe the behavior of f(x), substitute values to the left and right of the vertical asymptote. So, let x = −0.5, then f(x) is a positive number, so Next, let x = −1.5, then f(x) is a negative number, so

Example 7: Right-end behavior models give us: dominant terms in numerator and denominator

End Behavior Models: End behavior models model the behavior of a function as x approaches infinity or negative infinity. The function g is: a right end behavior model for f if and only if a left end behavior model for f if and only if

Example 8: (The x term dominates.) As , approaches zero. becomes a right-end behavior model. Test of model Our model is correct. As , increases faster than x decreases, therefore is dominant. becomes a left-end behavior model. Test of model Our model is correct.

p Example 8 (cont.): becomes a right-end behavior model. becomes a left-end behavior model. On your calculator, graph: Use: p