Question

Given that limit exists find \(lt_{(x,y,z)\rightarrow(2,2,2)}\left (\frac{ln(1+\frac{xy-2x-y+z}{xz-2x-6z+12}+\frac{xz-5x-2z+10}{xy-7y-2x+14}}{(x-2)(y-2)(z-2)}\right )\)

(a) ∞

(b) 1

(c) 0

(d) ln(^4⁄5)

This question was posed to me in an online interview.

Enquiry is from Limits and Derivatives of Several Variables in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct answer is (a) ∞

Explanation: We can parameterize the curve by

x = y = z = t

\(lt_{t\rightarrow 2}\left (\frac{ln(1+\frac{xy-2x-y+z}{xz-2x-6z+12}+\frac{xz-5x-2z+10}{xy-7y-2x+14}}{(x-2)(y-2)(z-2)}\right )\)

\(=lt_{t\rightarrow 2}\left (\frac{ln(1+\frac{t}{(t-6)}+\frac{(t-5)}{(t-7)}}{(t-2)^3}\right )\)

\(=lt_{t\rightarrow 2}\left (\frac{ln(1+\frac{2}{(t-6)}+\frac{(2-5)}{(2-7)})}{(2-2)^3}\right )=\frac{ln(\frac{4}{5})}{0}\rightarrow\infty\)