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PROBLEM:

Find the value or values of x for which the function is discontinuous.

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Solution:

A function $\displaystyle f\left( x \right)$ is continuous at $\displaystyle x=a$ if $\displaystyle \lim_{x \to a} f\left( x \right)=f\left( a \right)$, which implies these three conditions:

1. $\displaystyle f\left( a \right)$ is defined.
2. $\displaystyle \lim_{x \to a} f\left( x \right)=L$ exists, and
3. $\displaystyle L=f\left( a \right)$

We are given a rational function. A rational function is not defined when the denominator is equal to zero. If we equate the denominator to zero, we can compute the value/s of $\displaystyle x$ where the function is discontinuous.

The function is not continuous at $\displaystyle x=5$.

The graph of the function $\displaystyle f\left( x \right)=\frac{3x}{x-5}$ is drawn below. It can be seen that there is an infinite discontinuity at $\displaystyle x=5$.

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