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lim f (x) = +
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Fig. 9-3 When f (x) has an in nite limit as x approaches a from the right and/or from the left, the graph of the function gets closer and closer to the vertical line x = a as x approaches a. In such a case, the line x = a is called a vertical asymptote of the graph. In Fig. 9-4, the lines x = a and x = b are vertical asymptotes (approached on one side only).
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[CHAP. 9
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If a function is expressed as a quotient, F(x)/G(x), the existence of a vertical asymptote x = a is usually signaled by the fact that G(a) = 0 [except when F(a) = 0 also holds]. EXAMPLE Let f (x) =
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x 3+
x 2 = + x 3
x 3
x 2 = x 3
In this case, the asymptote x = 3 is approached from both the right and the left (see Fig. 9-5). 1 x 2 =1+ . Thus, the graph of f (x) is obtained by shifting the hyperbola y = 1/x three units to [Notice that, by division, x 3 x 3 the right and one unit up.]
Fig. 9-5
LIMITS AT INFINITY: HORIZONTAL ASYMPTOTES
As x gets larger without bound, the value f (x) of a function f may approach a xed real number c. In that case, we shall write
x +
lim f (x) = c
In such a case, the graph of f gets closer and closer to the horizontal line y = c as x gets larger and larger. Then the line y = c is called a horizontal asymptote of the graph more exactly, a horizontal asymptote to the right. EXAMPLE Consider the function f (x) =
a horizontal asymptote to the right. x 2 whose graph is shown in Fig. 9-5. Then lim f (x) = 1 and the line y = 1 is x + x 3
If f (x) approaches a xed real number c as x gets smaller without bound,1 we shall write
x
lim f (x) = c
1 To say that x gets smaller without bound means that x eventually becomes smaller than any negative number. Of course, in that case, the absolute value |x| becomes larger without bound.
CHAP. 9]
SPECIAL LIMITS
In such a case, the graph of f gets closer and closer to the horizontal line y = c as x gets smaller and smaller. Then the line y = c is called a horizontal asymptote to the left. EXAMPLES For the function graphed in Fig. 9-5, the line y = 1 is a horizontal asymptote both to the left and to the right. For the function graphed in Fig. 9-2, the line y = 0 (the x-axis) is a horizontal asymptote both to the left and to the right. If a function f becomes larger without bound as its argument x increases without bound, we shall write lim f (x) = + .
x +
EXAMPLES
x +
lim (2x + 1) = +
x +
lim x 3 = +
x
If a function f becomes larger without bound as its argument x decreases without bound, we shall write lim f (x) = + .
EXAMPLES
x
lim x 2 = +
x
lim x = +
If a function f decreases without bound as its argument x increases without bound, we shall write lim f (x) = .
x +
EXAMPLES
x +
lim 2x =
x +
lim (1 x 2 ) =
If a function f decreases without bound as its argument x decreases without bound, we shall write lim f (x) = .
x
EXAMPLES
x
lim 2x =
x
lim x 3 =
EXAMPLE Consider the function f such that f (x) = x [x] for all x. For each integer n, as x increases from n up to but not including n + 1, the value of f (x) increases from 0 up to but not including 1. Thus, the graph consists of a sequence of line segments, as shown in Fig. 9-6. Then lim f (x) is unde ned, since the value f (x) neither approaches a xed limit nor does it become larger
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