Question

Two men on a 3-D surface want to meet each other. The surface is given by \(f(x,y)=\frac{x^6.y^7}{x^{13}+y^{13}}\). They make their move horizontally or vertically with the X-Y plane as their reference. It was observed that one man was initially at (400, 1600) and the other at (897, 897). Their meet point is decided as (0, 0). Given that they travel in straight lines, will they meet?

(a) They will meet

(b) They will not meet

(c) They meet with probability ^1⁄2

(d) Insufficient information

The question was posed to me in an international level competition.

This intriguing question originated from Limits and Derivatives of Several Variables in portion Partial Differentiation of Engineering Mathematics

Answer 1

The correct answer is (b) They will not meet

The explanation is: The problem asks us to find the limit of the function f(x, y) along two lines y = x and y = 4x

For the first line (first person)

x = t : y = 4t

=\(lt_{t\rightarrow 0}\frac{t^6.4^7.t^7}{t^{13}+4^{13}t^{13}}=lt_{t\rightarrow 0}\frac{4^7t}{t(1+4^{13})}\)

=\(\frac{4^7}{1+4^{13}}\)

For the second line (Second Person)

 x = t = y

=\(lt_{t\rightarrow 0}\frac{t^6.t^7}{t^{13}+t^{13}}=lt_{t\rightarrow 0}\frac{t^{13}}{2t^{13}}\)

= ^1⁄2

The limits are different and the will not meet.