Question

Can someone explain to me how to tell if the information given to me is either a removable or non-removable discontinuity? please and thanks.

Answer 1

I'm going to assume that so far, you've only had to deal with rational polynomial functions \(R(x)=\dfrac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomials.

A removable discontinuity is when you have a value \(x=x_0\) such that \(Q(x_0)=0\), but you can factorize \(P(x)\) so that the factor with the discontinuity in the denominator can be removed.

For example, take \(R(x)=\dfrac{P(x)}{Q(x)}=\dfrac{x^2-1}{x-1}\), which has a discontinuity at \(x=1\). The numerator can be factored:
\[R(x)=\frac{(x+1)(x-1)}{x-1}=\begin{cases}x+1&\text{for }x\not=1\\\text{undefined}&\text{for }x=1\end{cases}\]
Note that this implies that \(\dfrac{x^2-1}{x-1}\not=x+1\). The reason for the different cases is that when \(x=1\), you would have the indeterminate expression \(\dfrac{0}{0}\), which means the function is undefined at this particular point.

A non-removable discontinuity is a point \(x=x_0\) that makes \(Q(x_0)=0\), but cannot be removed as above.

For example, consider \(R(x)=\dfrac{1}{x}\), which has a discontinuity at \(x=0\). There's no way to factorize the numerator for there to be a factor of \(x\) to eliminate the \(x\) in the denominator.

Answer 2

Thank you for the help.