We have to find the limit of the given function at the given point.

\(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}\)

\(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}\)

\(\Rightarrow\frac{0\ln0}{1}\)

now as \(\ln0=1\)

therefore

=0

i.e. \(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}=0\)

\(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}\)

\(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}\)

\(\Rightarrow\frac{0\ln0}{1}\)

now as \(\ln0=1\)

therefore

=0

i.e. \(\lim_{(x,y)\rightarrow(1,0)}\frac{y\ln y}{x}=0\)