Question

Find \(lt_{(x,y)\rightarrow(0,0)}\frac{y^6}{x^{10}y^2+x^{15}}\)

(a) 0

(b) 1

(c) Does Not exist

(d) ∞

This question was addressed to me in class test.

I'd like to ask this question from Limits and Derivatives of Several Variables in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct option is (c) Does Not exist

Explanation: Put Put x = t : y = a.t^^5⁄2 we have

=\(lt_{(x,y)\rightarrow(0,0)}\frac{(a.t^{\frac{5}{2}})^6}{t^{10}.(a.t^{\frac{5}{2}})^2+t^{15}}\)

=\(lt_{(x,y)\rightarrow(0,0)}\frac{a^6}{a^2+1}\)

By varying a we get different limits along different paths

Hence, Does Not exist is the right answer.