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

1 Answer
Jan 10, 2016

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.