For this problem you need to recognize that for a function to have a limit that exists, the limit just below your value (2) must be equal to the limit just above your value. When we plug your given value of 2 into the function this is what you get:
8(2)-11<= f(x) <= 2(2)^2-3
16-11 <= f(x) <= 2(4)-3
5 <= f(x) <= 8-3
5 <= f(x) <= 5
Since the limit value on the left and right of your inequality are equal your solution is 5.