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PROBLEM:
Find the value or values of x for which the function is discontinuous.
\large \displaystyle f\left( x \right)=\frac{3x}{x-5}
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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:
- \displaystyle f\left( a \right) is defined.
- \displaystyle \lim_{x \to a} f\left( x \right)=L exists, and
- \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.
\begin{align*} x-5 & = 0 \\ x & = 5 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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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