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[CHAP. 7
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7.24 Express the set of solutions of each inequality below in terms of the notation for intervals: (a) 2x + 3 < 9 (e) 3 < 4x 5 < 7 2x 5 <1 (i) x 2 (b) 5x + 1 6 (f ) 1 2x + 5 < 9 (j) x 2 6 (c) 3x + 4 5 (g) |x + 1| < 2 (k) (x 3)(x + 1) < 0 (d) 7x 2 > 8 (h) |3x 4| 5
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7.25 For what values of x are the graphs of (a) f (x) = (x 1)(x + 2) and (b) f (x) = x(x 1)(x + 2) above the x-axis GC Check your answers by means of a graphing calculator. 7.26 Prove Theorem 7.1. [Hint: Solve f (r) = 0 for a0 .] 7.27 Prove Theorem 7.2. [Hint: Make use of the algebra following Problem 7.21.]
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Limits
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8.1 INTRODUCTION To a great extent, calculus is the study of the rates at which quantities change. It will be necessary to see how the value f (x) of a function f behaves as the argument x approaches a given number. This leads to the idea of limit. EXAMPLE Consider the function f such that
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f (x) = x2 9 x 3
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whenever this formula makes sense. Thus, f is de ned for all x for which the denominator x 3 is not 0; that is, for x = 3. What happens to the value f (x) as x approaches 3 Well, x 2 approaches 9, and so x 2 9 approaches 0; moreover, x 3 approaches 0. x2 9 Since the numerator and the denominator both approach 0, it is not clear what happens to . x 3 However, upon factoring the numerator, we observe that x2 9 (x 3)(x + 3) =x+3 = x 3 x 3 Since x + 3 unquestionably approaches 6 as x approaches 3, we now know that our function approaches 6 as x approaches 3. The traditional mathematical way of expressing this fact is x2 9 =6 x 3 x 3 lim This is read: The limit of x2 9 as x approaches 3 is 6. x 3 Notice that there is no problem when x approaches any number other than 3. For instance, when x approaches 4, x 2 9 approaches 7 and x 3 approaches 1. Hence, 7 x2 9 = =7 1 x 4 x 3 lim
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PROPERTIES OF LIMITS
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In the foregoing example we assumed without explicit mention certain obvious properties of the notion of limit. Let us write them down explicitly.
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Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
LIMITS
[CHAP. 8
PROPERTY I.
lim x = a
This follows directly from the meaning of the limit concept. PROPERTY II. If c is a constant,
lim c = c
As x approaches a, the value of c remains c. PROPERTY III. If c is a constant and f is a function,
lim c f (x) = c lim f (x)
EXAMPLE
x 3 x 3
lim 5x = 5 lim x = 5 3 = 15
x 3 x 3 x 3
lim x = lim ( 1)x = ( 1) lim x = ( 1) 3 = 3
PROPERTY IV.
If f and g are functions,
lim [ f (x) g(x)] = lim f (x) lim g(x)
x a x a
The limit of a product is the product of the limits. EXAMPLE
lim x 2 = lim x lim x = a a = a2 x a x a x a More generally, for any positive integer n, lim x n = an . x a
PROPERTY V.
If f and g are functions,
lim [ f (x) g(x)] = lim f (x) lim g(x)
x a x a
The limit of a sum (difference) is the sum (difference) of the limits. EXAMPLES
(a) lim (3x 2 + 5x) = lim 3x 2 + lim 5x
x 2 x 2
= 3 lim x 2 + 5 lim x = 3(2)2 + 5(2) = 22
x 2 x 2
(b) More generally, if f (x) = an x n + an 1 x n 1 + + a0 is any polynomial function and k is any real number, then
lim f (x) = an k n + an 1 k n 1 + + a0 = f (k)
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