Question

Find the limit when X approaches zero
2xsinx/1-cosx

Answer 1

4

Explanation:

`Lim_(x \to 0)(2 x\sin x)/(1-\cos x)`

`=Lim_(x \to 0)(2 x\sin x)/(1-\cos x)* (1+\cos x)/(1+\cos x)`

`=Lim_(x \to 0)(2 x\sin x(1+\cos x) )/(1^2 -\cos^2 x)`

`=Lim_(x \to 0)(2 x\sin x(1+\cos x) )/(1 -\cos^2 x)`

`=Lim_(x \to 0)(2 x\sin x(1+\cos x) )/(sin^2 x)`

`=Lim_(x \to 0)(2x(1+\cos x) )/(sin x)`

`=Lim_(x \to 0) 2(1+\cos x) * (1)/(Lim_(x \to 0)(sin x)/(x))`

`=2(1+\cos 0) * 1`

`= 2(1+1)`

`= 2(2)`

`\therefore Lim_(x \to 0)(2 x\sin x)/(1-\cos x)= 4`