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Lecture 10

Lecture 10 - 2.6.pdf

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Mathematics & Statistics (Sci)
MATH 140
Ewa Duma

2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES In Sections 2.2 and 2.4 we investigated infinite limits and vertical asymptotes. There we let x approach a number and the result was that the values of y became arbitrarily large (positive or negative). In this section we let x become arbitrarily large (positive or nega- x f▯x▯ tive) and see what happens to y. 132 |||| CHAPTER 2 LIMITS AND DERIVATIVES 0 ▯1 Let’s begin by investigating the behavior of the function f defined by 2 ▯1 0 f▯x▯ ▯40 - Lecture 10 Notes Again, the symbol ▯▯ does not represent a number, but the expression lim f▯x▯ ▯ L ▯2 0.600000 x ▯ 1 y=ƒ is often read as ▯3 0.800000 as x becomes large. The table at the left gives values of this function correct to six decimal “the limit of f▯x▯, as x approaches negative infinity, is L” ▯4 0.882353 places, and the graph of f has been drawn by a computer in Figure 1. ▯5 0.923077 Limits at Infinity and Horizontal Asymptotes: ▯10 0.980198 y y=L Definition 2 is illustrated in Figure 3. Notice that the graph approaches the line y ▯ L as y=1 we look to the far left of each graph. ▯50 0.999200 0 x ▯100 0.999800 ▯1000 0.999998 3 y DEFINITION The line y ▯ L is called a horizontal asymptote of the curve 0 1 ▯-1 xy ▯ f▯x▯ if either y= ▯+1 FIGURE 1 lim f▯x▯ ▯ L y=ƒ xl▯ y=L As x grows larger and larger, the values of f(x) gets closer and closer to 1 but never to to 1. This implies that we have a horizontal asymptote at 1 which means that the value willing x sufficiently large. This situation is expressed symbolically by 0riting x For instance, the curve illustrated in Figure 1 has the line y ▯ 1 as a horizontal asymp- get extremely close to 1 but never 1 itself. tote because 2 lim x ▯FIGUR▯ 1 Definition: Let f be a function dx l ▯x ▯ 1ome interval (a, inf). Then lim (x > inf) f(x) = L Examples illustratixg_`im ƒ=L which means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. An example of a curve with two horizontal asymptotes is y ▯ tan x. (See Figure 4.) y In fact, ▯ Definition: Let f be a function defined on some interval (-inf, a). Then lim (x > -inf) f(x) = L which means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative. 0 4 lim tan x ▯ ▯ x xl▯▯ Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim ▯ (x > inf) f(x) = L or lim (x > -inf) f(x) = L _2 so both of the lines y ▯ ▯ ▯2 and y ▯ the fact that the lines x ▯ ▯▯2 are vertical asymptotes of the graph of tan.) Example: Find the infinite limits, limitsFIGURE 4ity, and asymptotes for the function f whose y=tan–!x EXAMPLE 1 Find the infinite limits, limits at infinity, and asymptotes for the function f graph is shown: whose graph is shown in Figure 5. y SOLUTIONWe see that the values of f▯x▯ become large as x l ▯1 from both sides, so 2 Notice that f▯x▯ becomes large negative as x approaches 2 from the left, but large posi- 0 2 x tive as x approaches 2 from the right. So lim f▯x▯ ▯ ▯▯ xl2▯ Thus both of the lines x ▯ ▯1 and x ▯ 2 are vertical asymptotes. We can see that y = inf when x approachFIGURE 5d when x approaches -1 from both As x becomes large, it appears that f▯x▯ approaches 4. But as x decreases through left and right side. Therefore the vertical asymptotes are when x = -1, 2.
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