Question

Precalculus: What is discontinuity in rational functions?

Is it another way of saying there is a hole in the graph?

Discontinuity: Anytime the denominator of a rational function equals zero.

What if it's 0/0? Is it a discontinuity or a hole? Or is it both?

Does it have to be a number over 0 to count as a discontinuity?

Answer 1

No, it is still a discontinuity.
Just imagine the function f(x) = x/x
does the point x=0 exist? Nope.

Answer 2

so 0/0 is not a hole?

Answer 3

It is a hole.
The numerator does no have to be over 0 to count as a discontinuity.

Answer 4

How does it count as a discontinuity?

Answer 5

You cannot divide by zero, therefore the point does not exist

Answer 6

it is a jump discontinuity

Answer 7

Wait, isn't that what a vertical asymptote is?

Answer 8

Do you have a graphing calculator?

Answer 9

or just a regular calculator?

Answer 10

graphing calculator

Answer 11

Graph the function y = x/x
then go to >2nd>calc>value
x=0

Answer 12

It will give you an error because the point does not exist, it is 0/0. And it is a jump discontinuity.
The condition for the discontinuity is that the denominator is 0, the numerator DOES NOT matter

Answer 13

You are confusing discontinuities with limits, where 0/0 could have an actual value (limit)

Answer 14

i guess you wanto knw how to knw whether a discontinuity has vertical asymptote or not

Answer 15

Wait, so in f(x) = 1/(3x^2+3x-18)
The vertical asymptotes are 2,-3 which are also the discontinuities?

Answer 16

x=2 and x=-3

Answer 17

are vertical asymptotes and discontinuities the same thing?

Answer 18

vertical asymptotes are discontinuities but discontinuities are not necessarily vertical asymptotes

Answer 19

Understand?

Answer 20

Why can't discontiunities be vertical aympsotes?

Answer 21

sometimes the factors leading to 1/0 cancel out in numerator and denum (just the case with y=x/x) leaving a discontinuity(hole) without any asymptote, cuz around the hole, graph stays at fixed value 1 here, it doesnt increase forever...

Answer 22

consider another example where factors leading to 1/0 doesnt cancel out.

y = 1/x