Chapter 5.6, Problem 103E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Continuous Function In Exercises 103 and 104, find the value of c that makes the function continuous at x = 0 f ( x ) = { 4 x − 2   sin   2 x 2 x 3 , x ≠ 0 c , x = 0

To determine

To calculate: The value of c that makes function f(x)={4x2sin2x2x3,x0c,x=0 continuous at x=0.

Explanation

Given:

The function is:

f(x)={4xâˆ’2sin2x2x3,â€‰â€‰â€‰xâ‰ 0c,â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰x=0.

And the limit of x tends to 0.

Formula used:

According to Lâ€²HÃ´pitalâ€²s rule if the limit of f(x)g(x) as x approaches to c produces indeterminate forms, then limxâ†’cf(x)g(x)=limxâ†’cfâ€²(x)gâ€²(x).

Calculation:

It is provided that the function f(x) is:

f(x)=4xâˆ’2sin2x2x3 â€¦...â€¦... (1)

Now put the limit x tends to 0 in equation (1) by using provided condition,

limxâ†’0f(x)=limxâ†’0(4xâˆ’2sin2x2x3) â€¦...â€¦... (2)

Now, as we put x equal to 0 in equation (2), it makes the form 00, which is an indeterminate form.

Therefore, apply Lâ€²HÃ´pitalâ€²s rule in equation (2),

limxâ†’0(4xâˆ’2sin2x2x3)=limxâ†’0ddx(4xâˆ’2sin2x)ddx(2x3)=limxâ†’0(4ddxxâˆ’2ddxsin2x)6x2=limxâ†’0(4âˆ’2Ã—2cos2x)6x2

Hence,

limxâ†’0ddx(4xâˆ’2sin2x)ddx(2x3)=limxâ†’04âˆ’4cos2x6x2 â€¦

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