Question

lim [(7-xy)/(x^2-3y^2)]
(x,y) --> (2,2)

Answer 1

\[\lim_{x,y \rightarrow 2,2} (7-xy)/(x^2+3y^2)\]

Answer 2

Since there is no discontinuity except at the origin you should be able to evaluate directly. However, in multiple dimensions the definition is a little bit more restrictive, namely:
\[\lim_{(x,y)\rightarrow (x_0,y_0)}\frac{f(x,y)-(f(x_0,y_0)+\sum_{i=1}^{2}\frac{df}{dx_i})}{\sqrt{(x-x_0)^2+(y-y_0))^2}}=0\]
All that definition means is that the distance between the point from all directions needs to go to zero SLOWER than the distance from the function to its linear approx goes to zero.

So, from every path the limit but be the same. Plugging in 2,2 you get:
\[\frac{7-(2)(2)}{16+48}=\frac{3}{64}\]