Question

Find \(lt_{(x,y)\rightarrow(0,0)}\frac{121.x^{-5}.y^{\frac{13}{3}}}{y+(x)\frac{3}{2}}\)

(a) ∞

(b) 0

(c) Does Not Exist

(d) 121

I got this question during an internship interview.

This intriguing question comes from Limits and Derivatives of Several Variables topic in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Correct choice is (c) Does Not Exist

To explain I would say: Put x = t : y = a.t^^3⁄2 we have

=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.t^{-5}.(at^{\frac{3}{2}})^{\frac{13}{3}}}{t^{\frac{3}{2}}+t^{\frac{3}{2}}}\)

=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.at^{\frac{13}{3}}.t^{\frac{3}{2}}}{2.t^{\frac{3}{2}}}\)

=\(lt_{(x,y)\rightarrow(0,0)}\frac{121.at^{\frac{13}{3}}}{2}\)

By varying a we get different limits along different paths

Hence, Does Not Exist is the right answer.