Answer
The limit does not exist.
Work Step by Step
We evaluate the limit
$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \ln \left( {3x + y} \right)$
along two different paths:
1. limit along the line $y=-3$
The limit becomes $\mathop {\lim }\limits_{x \to 1} \ln \left( {3x - 3} \right) = - \infty $
2. limit along the line $x=1$
The limit becomes $\mathop {\lim }\limits_{y \to - 3} \ln \left( {3 + y} \right) = - \infty $.
We conclude that the limit does not exist.
