Question

Determine the limit at the given point, if it exists. And is the function continuous at the given point?

f(x,y)=(x+1)y/1-x-y at the point (0,1)

Answer 1

This is what I did but I'm sure it's wrong.

\[f(x,y)=(x+1)y \div1-x-y \]
\[=xy+y \div 1-x-y\]
\[\lim_{(x,y) \rightarrow (0,1)} xy+y \div 1-x-y\]
\[=\lim_{(x,y) \rightarrow (0,1)} xy+y \div \lim_{(x,y) \rightarrow (0,1)} 1-x-y\]
\[=(\lim_{(x,y) \rightarrow (0,1)} x \times \lim_{(x,y) \rightarrow (0,1)} y + \lim_{(x,y) \rightarrow (0,1)} y) \div (1 \times \lim_{(x,y) \rightarrow (0,1)} -x -\lim_{(x,y) \rightarrow (0,1)} -y)\]
\[=xy+y \div 1(-x-y)\]
\[=xy+y \div -x-y\]
At the point (0,1):
\[=0 \times 1 + 1 \div -0 -1\]
\[=1 \div -1 = -1\]