Consider the following limit:

\(\lim_{(x,y,z)\rightarrow(0,1,0)}\ln(1+y)e^{xz}=\ln(1+1)e^{0\times0}\)

\(=\ln(2)e^0\)

\(=\ln(2)\times1\)

\(=\ln(2)\)

Hence, the solution is \(\lim_{(x,y,z)\rightarrow(0,1,0)}\ln(1+y)e^{xz}=\ln(2)\)

\(\lim_{(x,y,z)\rightarrow(0,1,0)}\ln(1+y)e^{xz}=\ln(1+1)e^{0\times0}\)

\(=\ln(2)e^0\)

\(=\ln(2)\times1\)

\(=\ln(2)\)

Hence, the solution is \(\lim_{(x,y,z)\rightarrow(0,1,0)}\ln(1+y)e^{xz}=\ln(2)\)