Explanation Given information: The function g ( x ) = { b if x ≤ 1 x − a x − 1 if x > 1 and f ( 11 ) = 11 . Condition for continuity: A function f ( x ) is continuous at a point x = a if and only if 1). lim x → a f ( x ) = f ( a ) is defined. 2). lim x → a − f ( x ) = f ( a ) is defined. 3). lim x → a + f ( x ) = f ( a ) is defined. Calculation: The rational function x − a x − 1 is undefined at x = 1 . So, the domain will be { x ∈ R | x ≥ 0 , x ≠ 1 } . The function g ( x ) is defined for all values of x. The only doubt point is x = 1 because g ( x ) is continuous for all non-negative real value. Check continuity at x = 1 . A function g ( x ) is continuous at a point x = 1 if and only if 1). lim x → 1 g ( x ) = g ( 1 ) lim x → 1 g ( x ) = b So, the function g ( 1 ) is defined. 2). When x approaches 1 from left hand side that means x should be x < 1 . So, the function will be g ( x ) = b lim x → 1 − g ( x ) = lim x → 1 − ( b ) lim x → 1 − g ( x ) = b 3). When x approaches 1 from right hand side that means x should be x > 1 . So, the function will be g ( x ) = x − a x − 1 . The limit, lim x → 1 + g ( x ) = lim x → 1 + x − a x − 1 lim x → 1 + g ( x ) = lim x → 1 + x − a ( x − 1 ) ( x + 1 ) [ ∵ ( x − 1 ) = ( x − 1 ) ( x + 1 ) ] If a = 1

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 2.3, Problem 40PS

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To find the constant a and b so that the given function will be continuous for all x.

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Chapter 2.3, Problem 39PS

Chapter 2.3, Problem 41PS

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Chapter 2 Solutions

Calculus

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To find the constant a and b so that the given function will be continuous for all x.

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Chapter 2 Solutions