Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {u,v} \right) \to \left( { - 1,0} \right)} \frac{{uv{e^{ - v}}}}{{{u^2} + {v^2}}} \cr
& {\text{Evaluating the limit}} \cr
& \mathop {\lim }\limits_{\left( {u,v} \right) \to \left( { - 1,0} \right)} \frac{{uv{e^{ - v}}}}{{{u^2} + {v^2}}} = \frac{{\left( { - 1} \right)\left( 0 \right){e^{ - 0}}}}{{{{\left( { - 1} \right)}^2} + {{\left( 0 \right)}^2}}} \cr
& {\text{Simplifying}} \cr
& \mathop {\lim }\limits_{\left( {u,v} \right) \to \left( { - 1,0} \right)} \frac{{uv{e^{ - v}}}}{{{u^2} + {v^2}}} = \frac{{\left( { - 1} \right)\left( 0 \right)\left( 1 \right)}}{{1 + 0}} \cr
& \mathop {\lim }\limits_{\left( {u,v} \right) \to \left( { - 1,0} \right)} \frac{{uv{e^{ - v}}}}{{{u^2} + {v^2}}} = 0 \cr} $$
