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If lim(xrarr0)(2a sin x-sin 2x)/(tan^(3)...

If `lim_(xrarr0)(2a sin x-sin 2x)/(tan^(3)x)` exists and is equal to 1, then the value of a is

A

2

B

1

C

0

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
`lim_(x to 0) (2 a sin x-sin2x)/(tan^(3)x)*(1)/(lim_(x to 0) ((sin x)/(x))^(3))(lim_(x to 0)cos^(3)x)`
`=lim_( x to 0) (2 a sin x -sin2x)/(x^(3)) [(0)/(0)"form"] = lim_(x to0)(2a cos x-2 cos 2x)/(3x^(2))` which will exist only if it is in `(0)/(0)` form.
`:.` We must have, `2 a cos theta -2 cos theta =0 rArr a=1`
Further, if a = 1, given limit becomes
`lim_(x to 0) (2 sin x -sin2x)/(tan^(3)x)=lim_(x to 0) (2 sin x (1-cos x))/(sin^(3)x).cos^(3)x=2*lim_(x to 0) ((1-cosx)/(x^(2)))*(1)/(lim_(x to 0) ((sin x)/(x))^(2))*lim_(x to 0) cos^(3)x=1`
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