Limits Involving Infinity Section 2.2 Limits Involving Infinity

Find the value of the following limits Determine your answer by analyzing the function. Then confirm it with tables and graphs. 1. lim 𝑥 → ∞ 2 −𝑥 2 𝑥 2. lim 𝑥 →−∞ 2 −𝑥 2 𝑥 3. lim 𝑥 →∞ sin 𝑥 2 𝑥 + cos 𝑥

AP Multiple Choice For which of the following does lim 𝑥 → ∞ 𝑓 𝑥 =0? I. 𝑓 𝑥 = ln 𝑥 𝑥 99 II. 𝑓 𝑥 = 𝑒 𝑥 ln 𝑥 III. 𝑓 𝑥 = 𝑥 99 𝑒 𝑥 A) I only B) II only C) III only D) I and II only E) I and III only

AP Mult Choice (To Boost Confidence) lim 𝑥 →2 𝑥 2 + 𝑥 − 6 𝑥 2 − 4 is A) -1/4 B) 0 C) 1 D) 5/4 E) Non-existent

Rational Function Theorem Given a rational function 𝑃(𝑥) 𝑄(𝑥) , where P(x) and Q(x) are polynomials. If degree of P(x) is < degree of Q(x), then…. lim 𝑥 → ±∞ 𝑃(𝑥) 𝑄(𝑥) = 0 (horizontal asymptote at 0) If degree of P(x) is > degree of Q(x), then…. lim 𝑥 → ±∞ 𝑃(𝑥) 𝑄(𝑥) = +∞ or −∞ If degree of P(x) is = degree of Q(x), then…. lim 𝑥 → ±∞ 𝑃(𝑥) 𝑄(𝑥) = leading coefficient of P(X) divided by the leading coefficient of Q(x). (horizontal asymptote)

End Behavior Model Notice that the Rational Function Theorem only considers the degree of the function. The other terms become insignificant as 𝑥 → ±∞. For example, 4x5 is an end behavior model for the function f(x) = 4x5 – 3x3 + 8x2 – 5 because they are nearly identical as x gets very large.

End Behavior Model Definition The function g is….. a right end behavior model for f if and only if lim 𝑥 → ∞ 𝑓(𝑥) 𝑔(𝑥) =1. a left end behavior model for f if and only if lim 𝑥 →−∞ 𝑓(𝑥) 𝑔(𝑥) =1. Ex: What are the right and left end behavior models for y = ex – 2x? Give examples of rational functions and find an end behavior model for them.

Vertical Asymptotes Although a limit does not exist as a real number, you can say that a limit approaches ±∞ at a vertical asymptote. Find the value for c where there is a vertical asymptote. Find the limits as x approaches c. 1. 𝑓 𝑥 = 𝑥 2 − 1 2𝑥+4 2. 𝑓 𝑥 = 1 𝑥 2

One More Method Notice that when x approaches infinity, 1/x approaches 0. Therefore, if it is easier, instead of analyzing f(x) as x →±∞, you can analyze f(1/x) as x approaches 0 (0+ or 0- for +∞ and -∞, respectively). 1. Find lim 𝑥→∞ 𝑥 sin 1 𝑥 .