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lim(x to 0)(int(-x)^(x) f(t)dt)/(int(0)^...

`lim_(x to 0)(int_(-x)^(x) f(t)dt)/(int_(0)^(2x) f(t+4)dt)` is equal to

A

f(0)

B

0

C

`(f(4))/(f(0))`

D

`(f(0))/(f(4))`

Text Solution

Verified by Experts

The correct Answer is:
D
`underset(x to 0)lim(underset(-x)overset(x)int f(t)dt)/(underset(0)overset(2x)int f(t+4)dt)`
`=underset(x to 0)lim((d)/(dx)(x)f(x)-(d)/(dx)(-x)f(-x))/((d)/(dx)(2x)f(2x+4)-0)`
`=underset(x to 0)lim(f(x)+f(-x))/(2f(2x+4))=(2f(0))/(2f(4))=(f(0))/(f(4))`
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