Answer to Question #288853 in Differential Equations for Pankaj

Question

If f(x, y) ={ 1 if x=0 or y=0 and 0 , otherwise } then lim f(x, y) does not exist for limit (x, y) approaches to (0, 0).

Answer 1

$\text{Consider the limit of f(x,y) along straight lines x$=$t, y$=$at, (or y$=$ax, where}\\ \text{a is the slope) as t}\rightarrow0^+. \text{We have,}\\ \lim_{(x,y) \rightarrow(0,0)}f(x,y)=\lim_{t \rightarrow0^+}f(t,at)=0, \text{if a}\neq0\ (\text{by definition of the function} ).\\ \text{And,}\\ \lim_{(x,y) \rightarrow(0,0)}f(x,y)=\lim_{t \rightarrow0^+}f(t,at)=1, \text{if a}=0.(\text{by definition of the function} )\\ \text{Thus, since the limit along a straight line depends on the slope of the line. We have}\\\text{that the two-variable limit does not exist.}$