Answer
The limit is
$$\lim_{(x,y,z)\to(-2,1,0)}xe^{yz}=-2.$$
The function is continuous everywhere.
Work Step by Step
Calculate this limit by substitution:
$$\lim_{(x,y,z)\to(-2,1,0)}xe^{yz}=-2e^{1\cdot0}=-2e^{0}=-2.$$
This function is continuous everywhere because at every point $(x_0,y_0,z_0)$ we can by substitution get
$$\lim_{(x,y,z)\to(x_0,y_0,z_0)}xe^{yz}=x_0e^{y_0z_0}.$$
