lim(x,y)-->(0,0) f(x,y) = 1 if y > (or equal to) |x| and 0 if y < |x|
I understand how to do a normal multivariable limit, but this problem is kind of bizarre. Maybe I'm just overthinking it? Thanks in advance for the help.
so your function says:
$\displaystyle
f\left( x,y \right) = \left\{ \begin{gathered}
1 \text{ } y \ge |x|\hfill \\
0 \text{ } y < |x|\hfill \\
\end{gathered} \right.
$
So if your looking at the case that f(0,0), y = x, and well if y = x your function goes to 1.
though x doesn't necessarily have to equal y. So you have to think about the 2 different cases.
If $\displaystyle x \ge y$ and you approach the origin what do you get, and if x < y and you approach the origin what do you get.
On the line y= 2x, x> 0; y= 2x> x= |x| so f(x,y)= 1 and the limit, as (x,y)goes to 0, on that line, is 1.
On the line y= x/2, x> 0; y= x/2< x= |x| so f(x,y)= 0 and the limit, as (x,y) goes to 0, on that line, is 0.
What does that tell you about "limit as (x, y) goes to (0, 0) of f(x,y)"?
(Unfortunately, seld is wrong. You cannot assume that y= x.)