Question

# Find the limit (if it exists) and discuss the continuity of the function. \lim_(x, y)\rightarrow (1, 1)\frac{xy}{x² + y²}

Limits and continuity
Find the limit (if it exists) and discuss the continuity of the function. $$\displaystyle\lim_{{{x},{y}}}\rightarrow{\left({1},{1}\right)}{\frac{{{x}{y}}}{{{x}²+{y}²}}}$$

2021-06-29
the function is continuous everywhere except the origin - the origin sets the denominator to 0
substitute (1,1) into the function, and we obtain the limit is $$\displaystyle{\frac{{{1}}}{{{2}}}}$$ - by continuity of the function.