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Evaluate lim(xto0) (2sinx-sin2x)/(x^(3))...

Evaluate `lim_(xto0) (2sinx-sin2x)/(x^(3))`

A

`0`

B

`1`

C

`2`

D

`3`

Text Solution

Verified by Experts

The correct Answer is:
B
Given, `underset(xto0)"lim"(2sinx-sin2x)/(x^(3)) = underset(xto0)"lim"(2sinx-2sinxcosx)/(x^(3))` `[therefore sin2x=2sinxcosx]`
`=underset(xto0)"lim"(2sinx(1-cosx))/(x^(3))`
`=2underset(xto0)"lim"(sinx)/(x). underset(xto0)"lim"(1-cosx)/x^(2)`
`[therefore underset(xto0)"lim"(sinx)/x=1]`
`=2underset(xto0)"lim"(1-1+sin^(2)x/2)/(x^(2)) = 2 underset(xto0)"lim"(2sin^(2)x/2)/(4 xx x^(2)/4)`
`=(2.2)/4 underset(xto0)"lim"((sinx/2)/(x/2))^(2)= underset(xto0)"lim"((sinx/2)/(x/2))^(2)=1`
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