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

Explanation

`lim_(x->oo) (sin(a+x) + sin(a-x) - 2 sin a)/(x sinx)`

`lim_(x->oo) (sina cosx +sinxcosa+ sina cosx -sinxcosa- 2 sin a)/(x sinx)`

`lim_(x->oo) (2sin a(cos x-1))/(x sinx)`

`=2sin a lim_(x to 0) (-2sin^2(x/2))/(x. 2sin(x/2)cos(x/2))`

`=-2sin a lim_(x to 0) (sin(x/2))/( 2(x/2)cos(x/2))`

`=-2sin a.(1/2)`

=`-sin a`