Question #288847

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).


1
Expert's answer
2022-01-31T16:00:56-0500

Consider the limit of f(x,y)f(x,y)  along straight lines x=t,y=at,(or y=axx=t, y=at, (or \ y=ax , wherea is the slope) as t0+t→0^+

.We have,

lim(x,y)(0,0)f(x,y)=limt0+f(t,at)=0,if a0lim_{(x,y)→(0,0)} ​f(x,y)=lim_{t→0^+} ​f(t,at)=0,if \ a \neq 0  (by definition of the function).And,

lim(x,y)(0,0)f(x,y)=limt0+f(t,at)=1,if a=0lim_{(x,y)→(0,0)} ​f(x,y)=lim_{t→0^+} ​f(t,at)=1,if \ a = 0(by definition of the function)Thus, since the limit along a straight line depends on the slope of te line. We have that the two-variable limit does not exist.


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