Ch. 2 – Limits and Continuity 2.2 – Limits Involving Infinity

Graph the following functions. Use the graph to find the desired limit. Horizontal asymptotes give you the limits at ±∞!

Rational Function Review Recall that rational functions (fractions with polynomials on top and bottom) often have horizontal asymptotes Ex: Find the horizontal asymptotes of the following functions. The degree of the top is smaller than the degree of the bottom Horizontal asymptote: y = 0 The degree of the top equals the degree of the bottom To find horizontal asymptote, divide leading coefficients: y = 3/2 The degree of the top is bigger than the degree of the bottom Divide top by bottom to get slant asymptote

Horizontal asymptotes give you the limits at ±∞! Ex: Find . Use what you know about horizontal asymptotes…think of the graph! Since f(x) has a vertical asymptote at y=0, f(x) will approach zero as x approaches infinity Answer = 0 Since f(x) has a slant asymptote at y=0, f(x) will approach infinity as x approaches infinity Answer = ∞ Horizontal asymptotes give you the limits at ±∞!

Random limit you must know: Some limits can be found as a result of several known limits! Ex: Find algebraically.

Ex: Find algebraically. Ex: Find philosophically. As x∞… …the numerator stays between -1 and 1. …the denominator becomes HUGE. What’s 1 divided by a huge number? Zero! So the answer is 0.

End Behavior Models Some complicated functions can be modeled with a simpler function at extreme values of x (±∞) Ex: Find power function end behavior models for the following functions: As x approaches ±∞, the -4x3 dominates the other terms, so we model this function with f(x) = -4x3 . As x approaches ±∞, the x2 terms dominate on top and bottom Since the x2 terms cancel out, we model this function with f(x) = 3 .

Ex: Find left and right end behavior models for the function y = x2 + e-x . Left-end behavior means as x-∞ Right-end behavior means as x∞ As x-∞, the e-x term dominates the function, so the left end model is f(x) = e-x . As x∞, the x2 term dominates the function (e-∞ is zero), so the right end model is f(x) = x2 . Left end model: Make a table to show that e^(-x) > x^2 for all x<0, including -inf