Answer
$$1$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \cos \left( {{x^2} + {y^2}} \right) \cr
& {\text{Rewrite the limit using polar coordinates}} \cr
& x = r\cos \theta ,{\text{ }}y = r\sin \theta \cr
& \left( {x,y} \right) \to \left( {0,0} \right),{\text{ so }}r \to 0 \cr
& {\text{Substituting}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \cos \left( {{x^2} + {y^2}} \right) = \mathop {\lim }\limits_{r \to 0} \cos \left[ {{{\left( {r\cos \theta } \right)}^2} + {{\left( {r\sin \theta } \right)}^2}} \right] \cr
& = \mathop {\lim }\limits_{r \to 0} \cos \left[ {{r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right)} \right] \cr
& = \mathop {\lim }\limits_{r \to 0} \cos \left( {{r^2}} \right) \cr
& {\text{Evaluate the limit when }}r \to 0 \cr
& = \cos \left( {{0^2}} \right) \cr
& = 1 \cr
& {\text{Then, we can conclude that}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \cos \left( {{x^2} + {y^2}} \right) = 1 \cr} $$
