How do I calculate this limit

How do I calculate the limit (as x approaches 3) of ((x+6)^0.5 - x) / (x^3 -3x^2)

Thanks in advance!

Rationalize the numerator to find the limit as under.

lim (x → 3) (√(x+6) - x) / (x^3 - 3x^2)
= lim (x → 3) [(√(x+6) - x) * (√(x+6) + x] / [x^2 (x - 3) * ((√(x+6) + x)]
= lim (x → 3) (x + 6 - x^2) / [x^2 (x - 3) * ((√(x+6) + x)]
= - lim (x → 3) [(x - 3) (x + 2)] / [x^2 (x - 3) * ((√(x+6) + x)]
= - lim (x → 3) (x + 2) / [x^2 * (√(x+6) + x)]
= - (3 + 2) / [3^2 * (√(3+6) + 3)]
= - 5/54.

limit (as x approaches 3) of ((x+6)^0.5 - x) / (x^3 -3x^2)

=limit (as x approaches 3) of {2-1/(x+6)^0.5}/(12x-6x^2)
=limit (as x approaches 3) of {2-1/(x+6)^0.5}/limit (as x approaches 3) of(12x-6x^2)
=-1/18 limit (as x approaches 3) of {2-1/(x+6)^0.5}
=1/18 limit (as x approaches 3) of {1/(x+6)^0.5-2}
=1/18{1/limit (as x approaches 3) of (1/(x+6)^0.5)-2}
=-5/54(Ans)

x->3 ((x+6)^(1/2)-x) *((x+6)^(1/2)+x)
----------------------------------------…
x^2(x-3) * ((x+6)^(1/2)+x)

x->3 --(x-3)(x+2)
---------------------
x^2(x-3)((x+6)^(1/2)+x)

--5/9x6=--5/54