Question

Find \(lt_{(x,y,z)\rightarrow(0,0,0)}\frac{y^2.z^2}{x^3+x^2.(y)^{\frac{4}{3}}+x^2.(z)^{\frac{4}{3}}}\)

(a) 1

(b) 0

(c) ∞

(d) Does Not Exist

I have been asked this question in homework.

I want to ask this question from Limits and Derivatives of Several Variables topic in chapter Partial Differentiation of Engineering Mathematics

Answer 1

Right answer is (d) Does Not Exist

Best explanation: Put x = t : y = a1 * t^^3⁄4 : z = a2 * t^^3⁄4

\(lt_{(x,y,z)\rightarrow(0,0,0)}\frac{(a_1)^2.t^{\frac{3}{2}}.(a_2)^2.t^{\frac{3}{2}}}{t^3+t^2.t.(a_1)^{\frac{4}{3}}+t^2.t.(a_2)^{\frac{4}{3}}}\)

\(lt_{(x,y,z)\rightarrow(0,0,0)}\frac{t^3}{t^3}\times \frac{(a_1)^2.(a_2)^2}{1+(a_1)^{\frac{4}{3}}+(a_2)^{\frac{4}{3}}}\)

\(lt_{(x,y,z)\rightarrow(0,0,0)} \frac{(a_1)^2.(a_2)^2}{1+(a_1)^{\frac{4}{3}}+(a_2)^{\frac{4}{3}}}\)

By varying a1 : a2  one can get different limit values.