Question

Find \(lt_{(x,y,z,w)\rightarrow(0,0,0,0)}\frac{x^{-6}.y^2.(z.w)^3}{x+y^2+z-w}\)

(a) 1990

(b) ∞

(c) Does Not Exist

(d) 0

The question was posed to me during an interview.

The above asked question is from Limits and Derivatives of Several Variables topic in section Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (c) Does Not Exist

Explanation: Put x = t : y = a1.t^^1⁄2 : z = a2.t : w = a3.t

\(lt_{(x,y,z,w)\rightarrow(0,0,0,0)}\frac{t^{-6}.t.(a_1)^2.t^6.(a_2)^3.(a_3)^3}{t+t.(a_1)^2+a_2.t-a_3.t}\)

\(lt_{(x,y,z,w)\rightarrow(0,0,0,0)}\frac{t}{t}\times \frac{(a_1)^2.(a_2)^3.(a_3)^3}{1+(a_1)^2+a_2-a_3}\)

\(lt_{(x,y,z,w)\rightarrow(0,0,0,0)} \frac{(a_1)^2.(a_2)^3.(a_3)^3}{1+(a_1)^2+a_2-a_3}\)

By changing the values of a1 : a2 : a3 we get different values of limit.

Hence, Does Not Exist is the right answer.