f(x) = |x^2 - 4|. Find the two points at which the function is not differentiable. For the larger of the two points, find the left andright limit of the slope of the function.

Expert Answers
mlehuzzah eNotes educator| Certified Educator

`|x^2-4|` is either `x^2-4` or `-(x^2-4)`

These are both differentiable functions (all polynomials are).  Thus the only time a problem can occur is when the function switches between `x^2-4` and `-(x^2-4)` .  That is, the only place the function can fail to be differentiable is when `|x^2-4| = 0`     

`|x^2-4|=0` can only happen if `x^2-4=0`

Thus: `(x-2)(x+2)=0` , so `x= +- 2`


Consider the larger of these two points, `x=2`

If x is a little bit larger than 2, then `x^2-4>0` , so `|x^2-4|=x^2-4`

But if x is a little bit smaller than 2, then `x^2-4<0` so `|x^2-4|=-x^2+4`

So we can take the right and left handed limits:

`lim_(x->2^+) (|x^2-4|-0)/(x-2)`

` = lim_(x->2^+) (x^2-4)/(x-2)`

` = lim_(x->2^+) ((x+2)(x-2))/(x-2)`

` = lim_(x->2^+) (x+2) = 4`

`lim_(x->2^-) (|x^2-4|-0)/(x-2)`

` = lim_(x->2^-) (-x^2+4)/(x-2)`

` = lim_(x->2^-) (-(x+2)(x-2))/(x-2)`

` = lim_(x->2^-) -(x+2) = -4`