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

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Answer 1 · 2018-06-28T10:03:47

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

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#