Evaluate the following limit: `lim_(x->0)(t a n x-sin x)/(sin3x-x sin x)`

Explanation

`lim_(x->0)(t a n x-sin x)/(sin3x-x sin x)`

`=lim_(x->0)((sin x)/(cos x)-sin x)/(3sinx- 4sin^3 x-3sin x)`

`=lim_(x to 0) (sin x(1/(cos x)-1))/(-4sin^3 x)`

`=-1/4 lim_(x to 0) (1/(cos x)-1)/(sin^2 x)`

`=-1/4 lim_(x to 0) ((1-cosx)/(cos x))/(1-cos^2x)`

`=-1/4 lim_(x to 0) (1-cos x)/(cosx(1-cosx)(1+cosx))`

`=-1/4 lim_(x to 0) 1/(cos x(1+cos x))`

`-1/4 xx1/(cos 0(1+cos 0))`

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