2. (2 points) Consider the piecewise defined function,f (x) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩4 + x if x < 29 if x = 212x if x > 2Evaluate limx→2(2f (x)).A. 6B. 18C. 12D. 9E. The limit does not exist

To solve this problem, we need to evaluate the limit of the function 2f(x) as x approaches 2.

The function f(x) is defined differently for values of x that are less than 2, equal to 2, and greater than 2.

1. If x < 2, f(x) = 4 + x. So, 2f(x) = 2(4 + x) = 8 + 2x.
2. If x = 2, f(x) = 9. So, 2f(x) = 2*9 = 18.
3. If x > 2, f(x) = 12x. So, 2f(x) = 2*12x = 24x.

Now, we need to evaluate the limit of 2f(x) as x approaches 2 from the left side (x < 2), from the right side (x > 2), and at x = 2.

1. The limit as x approaches 2 from the left side is lim(x->2-) 2f(x) = 8 + 2*2 = 12.
2. The limit as x approaches 2 from the right side is lim(x->2+) 2f(x) = 24*2 = 48.
3. At x = 2, 2f(x) = 18.

Since the left-hand limit (12) does not equal the right-hand limit (48), the limit of 2f(x) as x approaches 2 does not exist.

So, the answer is E. The limit does not exist.