There is nothing about the definition of q(x) that keeps you from evaluating the limit by substituting the value b for x in the appropriate definition.
a) from the left: lim = a(b -b)^3 +c = 0 + c = c
b) from the right: lim = a(b -b) +c = 0 + c = c
c) the limit from the left is equal to the limit from the right, so the limit exists and is that value: c.
d) the function is undefined at x=b, so q(b) does not exist. (The graph has a hole at that point.)
