Use series to evaluate the limit.

lim (x->0) (1-cos(3x))/(1+3x-e^(3x))

Using the series for sine and exp,
lim(x→0) [1 - cos(3x)] / [1 + 3x - e^(3x)]
= lim(x→0) [1 - (1 - (3x)^2/2! + (3x)^4/4! - ...)] / [1 + 3x - (1 + 3x + (3x)^2/2! + (3x)^3/3! + ...)]
= lim(x→0) [(3x)^2/2! - (3x)^4/4! + ...] / [-(3x)^2/2! - (3x)^3/3! + ...]
= lim(x→0) [3^2/2! - 3^4 x^2/4! + ...] / [-3^2/2! - 3^3 x/3! + ...]
= [3^2/2! - 0] / [-3^2/2! - 0]
= -1.

I hope this helps!