Question

Given that limit exists \(lt_{(x,y,z)\rightarrow(0,0,0)}\left (\frac{cos(\frac{\pi}{2}-x).tan(y).cot(\frac{\pi}{2}-z)}{sin(x).sin(y).sin(z)}\right )\)

(a) 99

(b) 0

(c) 1

(d) 100

This question was posed to me in homework.

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

Answer 1

Right choice is (c) 1

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

\(lt_{t\rightarrow 0}\left (\frac{cos(\frac{\pi}{2}-t).tan(y).cot(\frac{\pi}{2}-t)}{sin(x).sin(y).sin(z)}\right)\)

\(lt_{t\rightarrow 0}\frac{(sin(t))(tan^2(t))}{sin^3(t)}\)

\(=lt_{t\rightarrow 0}\frac{tan^2(t)}{sin(t)} = lt_{t\rightarrow 0}\frac{1}{cos^2(t)}\)

\(=\frac{1}{cos^2(0)}=\frac{1}{1}=1\)