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

Does the function # f(x) = x/(|x|-3)# have any discontinuities?

Answer 1

There are discontinuities when #x=+-3#

We have:

# f(x) = x/(|x|-3) #

Her is a graph of the function. At first appearance, it would look as through #f(x)# has discontinuities when #x=+-3# graph{x/(|x|-3) [-10, 10, -10, 10]}

Both the numerator and denominator are continuous functions in their own right. The combination leading to the definition of #f(x)# can therefore only have discontinuity when the denominator is zero.

ie there could only be a discontinuity if

# |x|-3 = 0 => |x| = 3 => x=+-3#

Hence, There are discontinuities when #x=+-3#

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Answer 2

Charles Anctil

Yes, the function f(x) = x/(|x|-3) has a discontinuity at x = 3.

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Answer 3

Adam Adkins

Yes, the function ( f(x) = \frac{x}{|x|-3} ) has a discontinuity at ( x = 3 ) because the denominator becomes zero at that point, which results in division by zero, making the function undefined. Hence, ( x = 3 ) is a point of discontinuity for this function.

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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.