Question

The function f(x, y)=\(\frac{x^2(2xy-3x)+9y}{3x-2y}\) is which of the following?

(a) Discontinuous at origin

(b) Discontinuous at (1, 1)

(c) Continuous at origin

(d) Discontinuous at (2, 2)

This question was posed to me during a job interview.

Question is taken from Continuity in division Complex Function Theory of Engineering Mathematics

Answer 1

The correct answer is (a) Discontinuous at origin

Explanation: Origin implies the coordinates (0, 0)

Case 1: \(\lim_{x \to 0 \\ y \to 0}\frac{x^2(2xy-3x)+9y}{3x-2y}\)

\(\lim_{y \to 0}\frac{9y}{-2y}\)

=\(\frac{-9}{2}\)

Case 2: \(\lim_{y \to 0 \\ x \to 0}\frac{x^2(2xy-3x)+9y}{3x-2y}\)

\(\lim_{x \to 0}\frac{-3x^3}{3x}\)

=0

Since, both the limits are not equal, the function is not continuous at origin.