Given,

We have to evaluate the limit of \(\lim_{(x,y)\rightarrow(4,0)}x^2y\ln xy\)

Calculation,

Let, y=mx, then

\(\lim_{x\rightarrow4}x^2(mx)\ln x(mx)=\lim_{x\rightarrow4}mx^3\ln mx^2=64m\ln16m\)

Since the limit is path-dependent. Hence the given limit does not exist.

We have to evaluate the limit of \(\lim_{(x,y)\rightarrow(4,0)}x^2y\ln xy\)

Calculation,

Let, y=mx, then

\(\lim_{x\rightarrow4}x^2(mx)\ln x(mx)=\lim_{x\rightarrow4}mx^3\ln mx^2=64m\ln16m\)

Since the limit is path-dependent. Hence the given limit does not exist.