Question

Find \(lt_{(x,y,z)\rightarrow(0,0,0)}\frac{sin(x).sin(y)}{x.z}\)

(a) ∞

(b) ^1⁄3

(c) 1

(d) Does Not Exist

I got this question in my homework.

This question is from Limits and Derivatives of Several Variables topic in portion Partial Differentiation of Engineering Mathematics

Answer 1

The correct option is (d) Does Not Exist

To elaborate: Put x = t : y = at : z = t

=\(lt_{t\rightarrow 0}\frac{sin(t).sin(at)}{t^2}\)

=\(lt_{t\rightarrow 0}\frac{sin(t)}{t}\times (a) \times lt_{t\rightarrow 0}\frac{sin(at)}{at}\)

= (1) * (a) * (1) = a