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

What is the difference between a vertical asymptote and a removable discontinuity?

Answer 1

Isabella Ackerman

A removable discontinuity causes a hole in the graph.

#y = x^2/x# looks like the line #y=x#, except for the hole at #(0,0)#

#y = (x^3-2x)/(x-2)# looks like the parabola #y = x^2# except it has a hole at #(2,4)#

A vertical asymptote is a vertical line that a graph approaches, line the line #x=1# for the graph of #y=x/(x-1)#

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Answer 2

Scarlett Allison

A vertical asymptote is a vertical line that a graph approaches but never touches. It occurs when the function approaches infinity or negative infinity as the input approaches a certain value.

A removable discontinuity, also known as a removable singularity, is a point on the graph where the function is undefined or has a hole. It can be removed by redefining the function at that point.

In summary, the main difference is that a vertical asymptote is a line that the graph approaches but never touches, while a removable discontinuity is a point where the graph has a hole or is undefined, but can be fixed by redefining the 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.