Is this supposed to be a proof? It is almost correct, except where you apply the Squeeze Theorem. You need to show why the Squeeze Theorem applies. You have only part of the condition. You have $1 \le f(x) \le (x-2)^2+1$. Next, show $\lim_{x \to 2} 1 = \lim_{x \to 2} (x-2)^2+1$. Then, you can apply the Squeeze Theorem as you stated.