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.

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

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`|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`