How can I solve this limit? lim x---->0 (sin^3x) /(x-sinx)^3

1 Answer
Jun 28, 2018

lim_(x->0)(sin^3x)/(x-sinx)^3 = +oo

Explanation:

Divide and multiply by x^3:

(sin^3x)/(x-sinx)^3 = sin^3x/x^3 * x^3/(x-sinx)^3

(sin^3x)/(x-sinx)^3 = (sinx/x)^3 * (1/((x-sinx)/x))^3

(sin^3x)/(x-sinx)^3 = (sinx/x)^3 * (1/(1-sinx/x))^3

Using the well known trigonometric limit:

lim_(x->0) sinx/x =1

and then:

lim_(x->0) (sinx/x)^3 =1

On the other hand to evaluate:

lim_(x->0) (1/(1-sinx/x))^3

note that the numerator is finite and the denominator tends to zero and is always positive because sinx < x. Then:

lim_(x->0) (1/(1-sinx/x))^3 =+oo

and so:

lim_(x->0)(sin^3x)/(x-sinx)^3 = +oo