I'm going to guess you meant to put x+4 in parentheses in the denominator, so that this is
asking for the value of c equaling the limit as x goes to -4 of the expression (1/x - (1/(-4))/(x - (-4). This is the derivative of the function 1/x, taken at the point -4, which you can find using the power rule to be -1/16.
Another way to get the same answer, without using any calculus rules, is to write (1/x + 1/4)/(x+4) as ( [4+x]/4x) /(x+4) = 1/4x, which tends to same value as x approaches -4. So c = -1/16.
If you didn't mean to write (1/x + 1/4)/x + 4 as (1/x + 1/4)/(x+4), then no finesse is required to avoid dividing by zero, and the limit is obtained by substituting x = -4 into the expression [(1/x + 1/4)/x] + 4 yielding 0 + 4 = 4. In this case c = 4.
