If 2x-2¡Üf(x)¡Üx^2-2x+2, determine

limx-->2f(x)=
(Use squeeze theorem)

To find the limit of f(x) as x approaches 2 using the squeeze theorem, we need to determine the lower and upper bounds of f(x) within the given inequality.

Given: 2x - 2 ≤ f(x) ≤ x^2 - 2x + 2

First, let's find the limits of the lower bound and upper bound separately as x approaches 2.

1. Lower Bound:
Taking the limit of the lower bound, 2x - 2, as x approaches 2:
limx→2 (2x - 2) = 2(2) - 2 = 2

2. Upper Bound:
Taking the limit of the upper bound, x^2 - 2x + 2, as x approaches 2:
limx→2 (x^2 - 2x + 2) = (2^2) - 2(2) + 2 = 4 - 4 + 2 = 2

Now, we have the lower bound as 2 and the upper bound as 2. Since the lower bound and upper bound are both equal to 2, we can conclude that f(x) also approaches 2 as x approaches 2.

Therefore, limx→2 f(x) = 2.