Question

Find the points of discontinuity, if any for the function:
$f\left( x \right) = \dfrac{{{x^2} - 9}}{{\sin x - 9}}$

Answer 1

Hint: To find the points of discontinuity we need to factor both the numerator and denominator.

Complete step-by-step answer:
A point of discontinuity occurs when a number is a zero of both the numerator and denominator. So, we need to find a common zero of both numerator and denominator to find the point of discontinuity.

Given function $\dfrac{{{x^2} - 9}}{{\sin x - 9}}$

Now factoring numerator

$

   = \dfrac{{{x^2} - {3^2}}}{{\sin x - 9}} \\

   = \dfrac{{\left( {x + 3} \right)\left( {x - 3} \right)}}{{\sin x - 9}} \\

 $

Now $f\left( x \right)$ is discontinuous for zero in denominator

but$\sin x - 9 \ne 0$ for any values of $x$.

Thus there are no common zeros of the denominator and numerator.

So, the given function is continuous for all real values of $x$.

Note: Here, we can also directly check the denominator as it is not zero for any value, there will be no need to factorize the numerator. Also, a point of discontinuity can also be referred to as a break or hole in the graph of the function. Thus, we can get a question also asking for a hole in the plotting of the graph of the given function.