2.3. Infinite limits and limits at infinity 2.3 Infinite limits and limits at infinity Consider the function f (x) = x1 . What happens to f (x) as x → 0? 1 x f (x) x f (x) .5 1 2 -1 .1 .001 10 1000 - .5 −1 −2 -.1 -.001 −10 −1000 .0000001 1000000 -.0000001 −1000000 4 2 -4 2 -2 4 -2 -4 We invent new notation to describe this situation. We say lim+ x→0 1 =∞ x and lim− x→0 1 = −∞. x 43 2.3. Infinite limits and limits at infinity Formally: • to say lim+ f (x) = ∞ means that as x gets closer and closer to a from the right, x→a the numbers f (x) grow without bound. a • to say lim+ f (x) = −∞ means that as x gets closer and closer to a from the x→a right, the numbers f (x) become more and more negative without bound. a • to say lim− f (x) = ∞ means that as x gets closer and closer to a from the left, x→a the numbers f (x) grow without bound. a • to say lim− f (x) = −∞ means that as x gets closer and closer to a from the x→a left, the numbers f (x) become more and more negative without bound. a All these situations are called infinite limits. infinite limit is as follows: The graphical description of an 44 2.3. Infinite limits and limits at infinity Definition 2.3 If lim+ f (x) = ±∞ or lim− f (x) = ±∞, we say the vertical line x→a x→a x = a is a vertical asymptote (VA) for f (x). Example: x = 0 is a VA for f (x) = x1 . NOTE: ∞ is not a number. It is only a symbol. However, in the context of limits, ∞ can be manipulated in some ways as if it was a number (we’ll see how in Chapter 3). For now you should remember these facts: lim+ x→0 1 =∞ x lim− x→0 1 = −∞ x One infinite limit to memorize: lim ln x = −∞ x→0+ 1 1 e Other infinite limits are computed using techniques we will study later, using some rules of arithmetic with ∞. 45 2.3. Infinite limits and limits at infinity Limits at infinity We want to consider the values of f (x) when x gets larger and larger without bound. For example, suppose f (x) = x1 : x f (x) 1 10 1 10 1 10000 10100 1010000 1 10000 1 10100 1 1010000 We say x→∞ lim f (x) = L if • (heuristically) when x grows without bound, f (x) approaches L. • (graphically) the graph of f looks like L L or We say lim f (x) = L if x→−∞ • (heuristically) when x becomes more and more negative without bound, f (x) approaches L. • (graphically) the graph of f looks like L or L Definition 2.4 If x→∞ lim f (x) = L or lim f (x) = L, we say the horizontal line y = L x→−∞ is a horizontal asymptote (HA) for f (x). 46 2.3. Infinite limits and limits at infinity E XAMPLE Consider the following graph of some unknown function f : 8 7 6 5 4 3 2 1 -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 2 3 4 5 6 7 8 -2 -3 -4 -5 -6 Based on this graph, find the following: 1. lim f (x) x→∞ 2. lim f (x) x→−∞ 3. 4. lim f (x) x→−3+ lim f (x) x→−3− 5. lim f (x) x→−3 6. lim+ f (x) x→3 7. lim− f (x) x→3 8. lim f (x) x→3 9. the equation(s) of any vertical asymptote(s) of f 10. the equation(s) of any horizontal asymptote(s) of f 47