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Section 2.2a Limits Involving Infinity Note: “Infinity” does not represent a real number……however, 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… 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: 1 f x x Our new limits: 1 lim 0 x x 1 lim 0 x x 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 lim f x b x or Ex: Find any H.A. of the graph of lim f x b x f x 2 1 x lim f x lim f x 2 x x 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 lim f x b x or Ex: Find any H.A. of the graph of Investigate with both a graph and a table… lim f x b x f x x x 1 lim f x 1 lim f x 1 x x H.A.: y = –1, y = 1 2 Sandwich Theorem Revisited Find lim f x x for sin x f x x First, what do the graph and table suggest??? Confirm Analytically: 1 sin x 1 And by the Sandwich Theorem: For x > 0, we have 1 sin x 1 x x x sin x 1 1 0 lim lim lim 0 x x x x x x 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 Rewrite: Find 5 x sin x 5 x sin x lim lim x x x x x sin x lim 5 lim 5 0 5 x x x Limits Involving Infinity Sometimes, a function outgrows all bounds (either positive or negative) as x approaches a finite number a we write: lim f x x a or lim f x xa Think back to the reciprocal function: 1 lim x 0 x and 1 lim x0 x 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 lim f x xa or lim f x xa 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. 1 x 1 x f x 2 2 x 5 x 3 2 x 1 x 3 1 Now, check the graph! V.A.: x , x 3 2 lim f x x 0.5 lim f x x 0.5 lim f x x 3 lim f x x 3