Correct Answer - Option 3 : Discontinuous at exactly three point
Concept:
A function is written in the form of the ratio of two polynomial functions is called a rational function.
Rational functions are continuous at all the points except for the points where denominator becomes zero.
\(f(x)=\dfrac{P(x)}{Q(x)}\)
where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
f(x) will be discontinuous at points where Q(x) = 0.
Calculation:
Given:
\(f(x)=\dfrac{4-x^2}{4x-x^3}\)
This is a rational function, so it will be discontinuous at points where denominator becomes zero.
4x - x3 = 0
x(4 - x2) = 0
x(22 - x2) = 0
x(2 + x)(2 - x) = 0
x = 0, x = - 2 and x = 2
Hence the function \(f(x)=\dfrac{4-x^2}{4x-x^3}\) will be discontinuous at exactly three points 0, - 2 and 2.
