Evaluate lim h->0: Solving Absolute Value Homework

[bcaie16]

Homework Statement

Evaluate lim h->0 (f(4+h)-f(4))/h
given f(x)=|x-4|-4


2. The attempt at a solution
Not too sure how to do this with absolute values, I've tried just subbing in small values for h, which gave me -1, but as this is an online question for marks, I am not too sure. Any assistance would be greatly appreciated.

 

[danielu13]

Depending on where you are in your calculus, you may realize that this is the limit definition of a derivative. If so, the formula for the derivative of an absolute value is as follows:

$f'(x) = \frac{x}{|x|}*x'$

Then use u-substitution to solve this derivative.

However, if you do not know about derivatives, there is a little more work involved. You have to split your function into two parts:

$f(x) = -4*|x-4|$

$-> g(x) = -4*(x-4)$ and $-4*(4-x)$


This is done with rules you should have learned in earlier math classes about working with absolute value functions. At this point, you will then evaluate your same limit, this time using g(x), but do note that you will have to change f(4+h) to g(h-4) and f(4) to g(-4) when doing the second part of the piecewise function.