2.2 Limits Involving Infinity Name: ____________________ Objectives: Students will be able to compute limits involving infinity using various methods. Let f(x) = 1/x. Find lim f(x) and lim f(x). x -> ∞ x -> -∞ The line y = b is a ___________ __________ of the graph of a function y = f(x) if either lim f(x) = b or lim f(x) = b. x -> ∞ x -> -∞ Apr 158:34 PM Example Use graphs and tables to find lim f(x) , lim f(x), and x -> ∞ x -> -∞ 2 identify all horizontal asymptotes of f(x) = x/√x + 1. Apr 158:52 PM 1 The Sandwich Theorem also holds for limits as x -> ±∞. Example Find lim sinx x -> ∞ x Apr 159:02 PM The properties of limits also hold for limits as x -> ±∞. Example Find lim 5x + sinx . x Apr 159:03 PM 2 Let's revisit f(x) = 1/x and find lim f(x) and lim f(x). x -> 0+ x -> 0- The line x = a is a ________ _________ of the graph of a function y = f(x) if either lim f(x) = ±∞ or lim f(x) = ±∞. x -> a+ x -> a- Example a.) Find the V.A. of the graph of f(x). b.) Describe the behavior of f(x) to the left and right of each V.A. f(x) = (x2 - 1)/(2x + 4) Apr 159:12 PM End Behavior Models For numerically large values of x, we can sometimes model the behavior of a complicated function by a simpler one that acts virtually in the same way. Let f(x) = 3x4 - 2x3 + 3x2 -5x + 6 and g(x) = 3x4. Show that while f and g are quite different for numerically small values of x, they are virtually identical for x large. Apr 167:43 AM 3 End Behavior Model The function g is: a.) a right end behavior model for f if and only if lim f(x)/g(x) = 1. x -> ∞ b.) a left end behavior model for f if and only if lim f(x)/g(x) = 1. x -> -∞ Examples Find an end behavior model for: a.) f(x) = 2x5 +x4 - x2 + 1 b.) f(x) = 2x3 - x2 + x - 1 3x2 - 5x + 7 5x3 + x2 + x - 5 Apr 167:49 AM A function's right and left end behavior models need not be the same function. Example Find a.) a simple basic function as a right end behavior model and b.) a simple basic function as a left end behavior model. f(x) = x2 + e-x We can investigate the graph of y = f(x) as x -> ±∞ by investigating the graph of y = f(1/x) as x -> 0. Example Find lim sin(1/x). x -> ∞ Homework: Pages 76-77: 1-55 odd, 59-64 Apr 167:54 AM 4 Apr 188:32 AM 5

© Copyright 2021 Paperzz