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Evaluate the following limit: lim(x->oo)...

Evaluate the following limit: `lim_(x->oo) (sin(a+x) + sin(a-x) - 2 sin a)/(x sinx))`

Text Solution

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`lim_(x->0) (sin(a+x)+sin(a-x)-2sina)/(xsinx)`
`=lim_(x->0) (sinacosx+sinxcosa+sinacosx-sinxcosa-2sina)/(xsinx)`
`=lim_(x->0) (2sina(cosx-1))/(xsinx)`
`=2sina lim_(x->0) (-2sin^2(x/2))/(x*2sin(x/2)cos(x/2))`
`=-2sina lim_(x->0)(sin(x/2))/(2*x/2*cos(x/2))`
As, `lim_(x->0) sinx/x = 1 and lim_(x->0) cos(x/2) = cos 0^@ = 1` `=2sina*(1/(2*1)) = -sina`
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