Limits and Continuity Question!! Please Help!

Find the values of a and b that make f continuous everywhere:

f(x) =

(x^2 - 4)/(x - 2) if x < 2

ax^2 - bx + 3 if 2 <=x < 3

2x - a + b if x>=3

Thanks so much!!!! :)

We can rewrite this as
f(x) = x + 2 if x < 2
......= ax^2 - bx + 3 if 2 ≤ x < 3
......= 2x - a + b if x ≥ 3.

(For the first branch, since x ≠ 2, we have (x + 2)(x - 2)/(x - 2) = x + 2.)
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Since each branch is defined by polynomials, we only need to check the junction points.

(i) At x = 2:
2 + 2 = a * 2^2 - 2b + 3 [via one-sided limits at x = 2]
==> 4a - 2b = 1.

(ii) At x = 3:
a * 3^2 - 3b + 3 = 2 * 3 - a + b [via one-sided limits at x = 3]
==> 10a - 4b = 3

Solving for a and b yields a = 1/2 and b = 1/2.

I hope this helps!