Question

Observe the figure. It is given that the function has no limit as (x, y) → (0 ,0) along the paths given in the figure. Then which of the following could be f(x, y)

(a) \(f(x,y) = \frac{x^7.y^8}{(x+y)}\)

(b) f(x,y) = x^2y^7

(c) \(f(x,y) = \frac{xy^2}{(x^2+y^2)}\)

(d) \(f(x,y) = \frac{x^6.y^2}{(y^5+x^{10})}\)

I got this question during a job interview.

My question comes from Limits and Derivatives of Several Variables in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (d) \(f(x,y) = \frac{x^6.y^2}{(y^5+x^{10})}\)

The explanation: The curves in the given graph are parabolic and thus they can be parameterized by

x = t : y = at^2

Substituting in Option \(f(x,y) = \frac{x^6.y^2}{(y^5+x^{10})}\) we get

=\(lt_{t\rightarrow 0}\frac{t^6.a^2.t^4}{a^5.t^{10}+t^{10}}\)

\(lt_{t\rightarrow 0}\frac{t^{10}}{t^{10}}\times \frac{a^2}{a^5+1}\)

\(lt_{t\rightarrow 0} \frac{a^2}{a^5+1}\)

By varying a we get different limits