Question

Find \(lt_{(x,y)\rightarrow(0,0)}\frac{sin(sin(y))}{x^n}\)

(a) Does not Exist

(b) 0

(c) ∞

(d) 1

I had been asked this question in examination.

I'm obligated to ask this question of Limits and Derivatives of Several Variables in section Partial Differentiation of Engineering Mathematics

Answer 1

The correct answer is (a) Does not Exist

The explanation is: Put x = at : y = t

\(=lt_{t\rightarrow 0}\frac{1}{a^nt^{n-1}}\times \frac{sin(sin(t))}{t}\)

\(=lt_{t\rightarrow 0}\frac{1}{a^nt^{n-1}} \times (1)\)

By varying n we get different values of limits.