Question

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

(a) 0

(b) ∞

(c) 1

(d) Does Not Exist

The question was asked in homework.

My doubt stems from Limits and Derivatives of Several Variables in division Partial Differentiation of Engineering Mathematics

Answer 1

Right option is (d) Does Not Exist

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

\(=lt_{t\rightarrow 0}\frac{sin(t)}{a^n.t^n}\)

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

By varying n we get different limits

Hence, Does Not Exist is the right answer.