The concept at play here is that both limits represent the limit definition of the derivative at a specific point, in this case (8, 0). Although this absolute value function is continuous on the reals, it is NOT differentiable at x = 8 because the graph has a sharp corner. In other words, the instantaneous rate of change of f is undefined there because, as we will see, the left-handed and right-handed limits of the derivative are unequal.
Btw, if we look at the graph of f(x), it should be easy to see that the derivative = -1 for x < 8 and the derivative = 1 for x > 8 (since f(x) is piecewise linear, its derivative is piecewise constant). Indeed, these are the limits we will calculate below:
Because |x| flips the sign of x (i.e. |x| = -x) when x < 0 and does nothing to x (i.e. |x| = x) when x ≥ 0 , it follows that |x - 8| = 8 - x for x < 8 and |x - 8| = x - 8 for x ≥ 8. These are the definitions we use to calculate the limits.
limx→8- f(x) - f(8) / (x-8)
= limx→8- 8 - x - 0 / (x-8)
= limx→8- - 1 = -1
and
limx→8+ f(x) - f(8) / (x-8)
= limx→8+ x - 8 - 0 / (x-8)
= limx→8+ 1 = 1
