What are the removable and non-removable discontinuities, if any, of #f(x)=(x^2 - 3x + 2)/(x^2 - 1) #?

Answer 1

Removable at #x=1# and non-removable at #x=-1#

#f(x)=(x^2 - 3x + 2)/(x^2 - 1) # is a rational function.
Therefore, #f# is continuous on its domain.
The domain of #f# is #RR - {-1,1}#.
So #f# has discontinuities at #-1# and at #1#. (It is continuous everywhere else.)
In order to determine whether a discontinuity is removable, we need to look at the limit of #f# as #x# approaches the discontinuity.
At #x=1# #lim_(xrarr1)f(x) = lim_(xrarr1)(x^2 - 3x + 2)/(x^2 - 1)#.
This limit has indeterminate form #0/0#, so we factor and simplify.
#lim_(xrarr1)f(x) = lim_(xrarr1)((x-2)(x-1))/((x+1)(x-1)#
# = lim_(xrarr1)(x-2)/(x+1) = -1/2#

Because the limit exists, the discontinuity is removable.

At #x=-1# #lim_(xrarr-1)f(x) = lim_(xrarr1)(x^2 - 3x + 2)/(x^2 - 1)#.
This limit has form #6/0#, so the limit does not exist and the discontinuity cannot be removed.
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Answer 2

The function f(x)=(x^2 - 3x + 2)/(x^2 - 1) has a removable discontinuity at x = 1 and non-removable discontinuities at x = -1 and x = 1.

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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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