Question

Which of the following is discontinuous at x=9?

(a) \(\frac{x^2-81}{x-9}\)

(b) \(\frac{x^3-20x^2+117x-162}{x-9}\)

(c) \(\frac{x^2-11x+18}{x(x-9)}\)

(d) \(\frac{x^2-13x-90}{x-9}\)

I had been asked this question at a job interview.

My query is from Continuity in section Complex Function Theory of Engineering Mathematics

Answer 1

Correct option is (d) \(\frac{x^2-13x-90}{x-9}\)

For explanation: \(\frac{x^2-81}{x-9}\)=\(\frac{(x-9)(x+9)}{x-9}\)=(x-9) which is continuous at x = 9.

 \(\frac{x^3-20x^2+117x-162}{x-9}\)=\(\frac{(x-9)^2(x-2)}{(x-9)}\)=(x-9)(x-2) which is continuous at x = 9.

\(\frac{x^2-11x+18}{x(x-9)}\)=\(\frac{(x-2)(x-9)}{x(x-9)}\)=\(\frac{(x-2)}{x}\) which is continuous at x = 9.

\(\frac{x^2-13x-90}{x-9}\)=\(\frac{(x-18)(x+5)}{x-9}\) which is discontinuous at x = 9, since the denominator becomes 0.