For what value of k makes the limit exist

Question

lim→3 (x²–x+k) ⁄ (x–3)

Answer 1

We need to find such k that the following will hold

x²–x+k = (x–3)(ax+b)

x²–x+k = ax²+(b-3a)x-3b

From here it follows that (equate coefficients from both sides)

x²: a=1,

x: b-3a=-1

x⁰: k=-3b

==> a=1, b=2, k=-6

(x-3)(x+3)= x^2-x-6

So k=-6