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If L=lim(x->0)(sin2x+asinx)/(x^3) is f...

If `L=lim_(x->0)(sin2x+asinx)/(x^3)` is finite, then find the value of `a`

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`L= underset(xto0)lim(sin2x+asinx)/(x^(3))`
`underset(xto0)lim(2sinxcosx+asinx)/(x^(3))`
`=underset(xto0)lim(sinx)/(x)(2cosx+a)/(x^(2))`
`=underset(xto0)lim(2cosx+a)/(x^(2))`
Now, `D^( r )` tends to 0 when `xto0.` Then `N^( r )` also must tend to zero for which `underset(xto0)lim(2cosx+a)=0impliesa=-2.` Now,
`L= underset(xto0)lim(2cosx-2)/(x^(2))=-2underset(xto0)lim(2"sin"^(2)(x)/(2))/(x^(2))=-1`
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