Question

Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as (x, y)\rightarrow (0, 0). f(x, y) = 1 - \frac{\cos(x^{2}+y^

Composite functions
Discuss the continuity of the function and evaluate the limit of f(x, y) (if it exists) as $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}.{f{{\left({x},{y}\right)}}}={1}-{\frac{{{\cos{{\left({x}^{{{2}}}+{y}^{{{2}}}\right)}}}}}{{{x}^{{{2}}}+{y}^{{{2}}}}}}$$
the limit does not exist - one can consider $$\displaystyle{z}={x}^{{{2}}}+{y}^{{{2}}}$$, when $$\displaystyle{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)},{z}\rightarrow{0}^{{+}}$$, and $$\displaystyle{\cos{{\left({z}\right)}}}\rightarrow{1}$$, but the denominator goes to $$\displaystyle\infty$$, thus the limit does not exist.