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#

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.
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Answer 2

The function f(x) = abs(x-9)/ (x-9) has a removable discontinuity at x = 9. There are no non-removable discontinuities.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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