Home > Calculus > Classifying Topics of Discontinuity (removable vs. non-removable)

What is the discontinuity of the function #f(x) = |x-5|/(x-5)# ?

Answer 1

There is a jump discontinuity at #x=5#.

The signum function #sgn(x)=|x|/x# returns the sign of a number. For example, #sgn(-6)=-1# because the input was a negative number. The signum function described above has a jump discontinuity at 0 because a negative input returns -1 and a positive input returns +1. Therefore, #lim_(xrarr0^-)|x|/x=-1# and #lim_(xrarr0^+)|x|/x=+1#. The function #f(x)=|x-5|/(x-5)# is just the signum function shifted right 5 units, to the jump discontinuity of the signum function at #x=0# is also shifted right 5 units to #x=5#, so there is a jump discontinuity at #x=5#.

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Answer 2

Elijah Abram

The function f(x) = |x-5|/(x-5) has a discontinuity at x = 5.

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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.