Question

What are the removable and non-removable discontinuities, if any, of `f(x)=abs(x-9)/ (x-9)`?

Answer 1

There's a non-removable discontinuity in `x = 9`

Explanation:

Since we can't divide by `0` we know that `x != 9`, so that's a discontinuity.

It might look it's removable because the numerator has `|x-9|`

But, if you look at what happens before and after `x = 9` you'll see it's non-removable.

For `x < 9`

`y = |x-9|/(x-9) = -(x-9)/(x-9) = -1`

For `x > 9`

`y = |x-9|/(x-9) = (x-9)/(x-9) = 1`

So the graph still has a discontinuity, because the limit at `x = 9` doesn't exist.