Explanation Theorem used: Limits of the linear functions . Let a, b and m be real numbers. For linear functions f ( x ) = m x + b , lim x → a f ( x ) = f ( a ) = m a + b . Limit Laws: Suppose that c is a constant and the limits lim x → a f ( x ) and lim x → a g ( x ) exists, then, Limit law 1: lim x → a ( f ( x ) + g ( x ) ) = lim x → a f ( x ) + lim x → a g ( x ) Limit law 3: lim x → a ( c f ( x ) ) = c lim x → a f ( x ) Limit law 4: lim x → a ( f ( x ) g ( x ) ) = ( lim x → a f ( x ) ) ( lim x → a g ( x ) ) Limit law 5: lim x → a f ( x ) g ( x ) = lim x → a f ( x ) lim x → a g ( x ) Calculation: Consider the denominator, x − 16 (1) By using factorization formula, the equation (1) becomes, x − 16 = ( x 4 ) 4 − ( 2 ) 4 x − 16 = ( x 4 − 2 ) ( x 3 4 + 2 x 2 4 + 4 x 4 + 8 ) (2) Obtain the limit of the function by using the limit laws as shown below. lim x → 16 x 4 − 2 x − 16 = lim x → 16 ( x 4 − 2 ) lim x → 16 ( x − 16 ) ( by limit law 5 ) = lim x → 16 ( x 4 − 2 ) lim x → 16 ( ( x 4 − 2 ) ( x 3 4 + 2 x 2 4 + 4 x 4 + 8 ) ) ( by the equation 2 ) = lim x → 16 ( x 4 − 2 ) lim x → 16 ( x 4 − 2 ) lim x → 16 ( x 3 4 + 2 x 2 4 + 4 x 4 + 8 ) ( by limit law 4 ) = 1 lim x → 16 ( x 3 4 + 2 x 2 4 + 4 x 4 + 8 ) Simplify further and obtain the value of the limit as follows

Calculus: Early Transcendentals, 2nd Edition

2nd Edition

ISBN: 9780321965165

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Limits

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Chapter 2.3, Problem 76E

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To evaluate: The value of limx→16x4−2x−16 by using factorizing formula xn−an=(x−a)(xn−1+xn−2a+xn−3a2+⋯+xan−2+an−1) , where n is positive integer and a is a real number.

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Chapter 2.3, Problem 75E

Chapter 2.3, Problem 77E

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Chapter 2 Solutions

Calculus: Early Transcendentals, 2nd Edition

Ch. 2.1 - Suppose s(t) is the position of an object moving...Ch. 2.1 - Suppose s(t) is the position of an object moving...Ch. 2.1 - What is the slope of the secant Line that passes...Ch. 2.1 - Describe a process for finding the slope of the...Ch. 2.1 - Describe the parallels between finding the...Ch. 2.1 - Graph the parabola f(x) = x2. Explain why the...Ch. 2.1 - Basic Skills 7. Average velocity The function s(t)...Ch. 2.1 - Average velocity The function s(t) represents the...Ch. 2.1 - Average velocity The position of an object moving...Ch. 2.1 - Average velocity The position of an object moving...

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The value of lim x → 16 x 4 − 2 x − 16 by using factorizing formula x n − a n = ( x − a ) ( x n − 1 + x n − 2 a + x n − 3 a 2 + ⋯ + x a n − 2 + a n − 1 ) , where n is positive integer and a is a real number.

Solution Summary: The author evaluates the value of undersetxto 16mathrmlimsqrt by using a factorizing formula.

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To evaluate: The value of limx→16x4−2x−16 by using factorizing formula xn−an=(x−a)(xn−1+xn−2a+xn−3a2+⋯+xan−2+an−1) , where n is positive integer and a is a real number.

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Chapter 2 Solutions