Review: Limits of Functions - 10/7/16

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1 Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left hand limit of f() as approaches a. We write lim a + f() = L to mean that the function f() approaches L as approaches a from the right. We call this the right hand limit of f() as approaches a. Definition lim a f() = L lim a = lim a + = L. Eample Let the above graph be f(). Then lim 1 f() = 3, and lim 1 + f() = 5. Since lim 1 f() lim 1 + f(), the limit as approaches 1 does not eist. For 2, lim 2 f() = 7 = lim 2 + f(). Thus lim 2 f() = 7. Eample Let g() = +1. What is lim 1 g()? We want to evaluate some values of that are less than 1, but getting closer and closer to 1. g() It looks like as we re getting closer and closer to 1 from the left, the values of g() are getting closer and closer to 2, so we can guess that lim 1 = 2. Now what about lim 1 + g()? We do the same thing, but with values of that are bigger than 1. 1

2 g() It looks like as we re getting closer and closer to 1 from the right, the values of g() are also getting closer and closer to 2, so we can guess that lim 1 + = 2. Since we got the same values for both, we can guess that lim 1 g() = 2. 2 Discontinuities The limit lim a f() is not impacted by the value f(a), only by the values f() for the around a. Eample Let the following function be f(). Then what is lim 1 f()? We have lim 1 f() = lim 1 + f() = 1, so our overall limit is lim 1 f() = 1. Now let the following function be g(). What is lim 1 g()? We still have lim 1 g() = lim 1 + g() = 1, so our overall limit is still lim 1 = 1. 2

3 3 Vertical Asymptotes Definition We say that lim a f() = if f() can be made arbitrarily large by taking values close to a on both sides. We say that lim a f() = if f() can be made arbitrarily small by taking values close to a on both sides. 1 Eample lim 0. Let s use both of our techniques here - estimating numerically and 2 drawing a graph. If we estimate numerically, we can make a chart: f() ±.5 4 ± ±.01 10,000 ±.001 1,000,000 On either side of 0, when we plug in values of we just get bigger and bigger, so we guess that lim 0 f() =. We could also draw the graph: Eample Let g() = 1. Then let s look at the graph: lim 0 g() = and lim 0 + g() =. Thus the limit does not eist. Definition The line = a is a vertical asymptote if lim a + or lim a (or both) is infinite ( of ). Both of the above eamples have vertical asymptotes at zero, even though the limit itself doesn t eist for 1. We can also have only one side approaching infinity: 3

4 A rational function R() = P () has a vertical asymptote at a if Q(a) = 0 AND after Q() canceling terms with the numerator, a still gives zero in the denominator. Eample How many vertical asymptotes does g() = 2 1 have? What are they? We 2 2 start by seeing where the denominator is zero. 2 2 = ( 2)( + 1), so it is zero at = 1, 2. Can either of these factor with the numerator? The numerator factors as ( + 1)( 1), so the fraction can be written as (+1)(), so we could cancel out the + 1. Thus there is a hole at (+1)( 2) = 1. Can we cancel the 2? No, so this is a vertical asymptote. Thus we have one vertical asymptote at = 2. 4 Horizontal Asymptotes Definition The line y = L is a horizontal asymptote of f() if lim f() = L or lim f() = L. Eample lim arctan() = π/2, lim arctan() = π/2. Thus arctan() has horizontal asymptotes at π/2 and π/2. Say we have a function R() = P () Q(). If the degree of P is greater that the degree of Q (e.g. f() = 3 +3), then the function does 2 1 not have a horizontal asymptote. If the degree of P is less than the degree of Q (e.g. g() = 2 +7), then the function has a 5 4 horizontal asymptote at zero. If the degree of P equals the degree of Q (e.g. h() = 3+5 ), then the horizontal asymptote 7 is the ratio of the leading coefficients (here we would have a horizontal asymptote in both directions of 3). 4

5 Eample What is are the vertical and horizontal asymptotes of f() = +3? Let s 2 2 start with horizontal asymptotes. The bottom grows a lot faster than the top when I plug in large numbers, so lim f() = 0 = lim f(). Thus I have a horizontal asymptote at zero. For +3 vertical asymptotes, I can factor the bottom to give f() =. So is 3 and asymptote? No, (+3)( 4) it is a hole that can be filled in? Is 4 an asymptote? I can never get rid of the 4 on the bottom, so = 4 is a vertical asymptote. Practice Problems 1. What is lim 3? Either estimate numerically or use a graph (or both). 2. What is lim 2 1 1? Either estimate numerically or use a graph (or both). Is this an asymptote or a hole? 3. What are the horizontal and vertical asymptotes of ? 4. What are the horizontal and vertical asymptotes of ? 5. What are the horizontal and vertical asymptotes of 3? 6. What are the horizontal and vertical asymptotes of ? Solutions 1. lim 3 = lim = 2. This is a hole. 3. The horizontal asymptote is zero, and the vertical asymptote is = There are no horizontal or vertical asymptotes. 5. The horizontal asymptote is y = 1 3, and the vertical asymptote is = The horizontal asymptote is 1. The vertical asymptotes are = 2 and = 1. 5