Answer
1. Figure 7 (A):
Because we can approach $P$ from many different directions. Therefore, the limit $\mathop {\lim }\limits_{\left( {x,y} \right) \to P} f\left( {x,y} \right)$ does not exist
2. Figure 7 (B):
The point $Q$ is located at the level curve $g\left( {x,y} \right) = 4$. Since $g\left( {x,y} \right)$ is continuous, so
$\mathop {\lim }\limits_{\left( {x,y} \right) \to Q} g\left( {x,y} \right) = 4$
The limit exists.
Work Step by Step
From Figure 7 (A) we see that we can approach $P$ from many different directions. Therefore, limit $\mathop {\lim }\limits_{\left( {x,y} \right) \to P} f\left( {x,y} \right)$ does not exist.
From Figure 7 (B) it appears that $\mathop {\lim }\limits_{\left( {x,y} \right) \to Q} g\left( {x,y} \right)$ exists. We see that the contour interval is $2$. The point $Q$ is located at the level curve $g\left( {x,y} \right) = 4$. Since $g\left( {x,y} \right)$ is continuous, so
$\mathop {\lim }\limits_{\left( {x,y} \right) \to Q} g\left( {x,y} \right) = 4$
