I think that we have to look at one-sided limits any time the function behaves differently on different sides of the value in question.  Since the limit from the left is 2 but the limit from the right is 4, I think there is no single limit for x=-3 without specifying the direction, so choice D "The limit doesn't exist." would be correct.
(My calculus is very rusty.  I could be wrong.)
(I also know you posted this over a year ago, so you may not care any more.)

In piecewise-defined functions like this, we need to check both the left-hand limit and the right-hand limit.

If the two one-sided limits are equal, then the limit exists. If they are different, then the limit does not exist.

As x approaches -3 from the left, we use the first function, sqrt(7 + x).

Substituting x = -3 gives:
sqrt(7 - 3) = sqrt(4) = 2.

As x approaches -3 from the right, we use the second function, x^2 - 5.

Substituting x = -3 gives:
(-3)^2 - 5 = 9 - 5 = 4.

The left-hand limit is 2, and the right-hand limit is 4.

Since these values are not equal, the limit at x = -3 does not exist.

Therefore, the correct answer is D: The limit doesn't exist.