Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} {e^{ - 1/\left( {{x^2} + {y^2}} \right)}} \cr
& {\text{Using the limit laws }} \cr
& = {e^{\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \left[ { - 1/\left( {{x^2} + {y^2}} \right)} \right]}} \cr
& {\text{then}}{\text{, substituting 0 for }}x{\text{ and }}y \cr
& = {e^{ - 1/\left( {{{\left( 0 \right)}^2} + {{\left( 0 \right)}^2}} \right)}} \cr
& {\text{simplifying}} \cr
& = {e^{ - 1/0}} \cr
& = {e^{ - \infty }} \cr
& = 0 \cr} $$
