How to solve limit of (x^2 - 27)/(x-3) as x approaches 3.

is the limit exists?

The numearator is the difference of two cubes (I am assuming that you made a typo)

lim(x → 3) (x – 3)(x² + 3x + 9)/(x – 3) = lim(x → 3) (x² + 3x + 9 = 9 + 9 + 9 = 27

((x + 9) (x-3) / (x-3) = x+9 and now take the limit