consider the following functions: f(x,y)={x sin (1/y) + y sin(1/x), xy=/0 0, xy=
ID: 3343644 • Letter: C
Question
consider the following functions:
f(x,y)={x sin (1/y) + y sin(1/x), xy=/0
0, xy=0}
g(x,y)={(xy/x^2+y^2)+ y sin (1/x), x=/0
0, x=0}
h(x,y)= {(xy/x^2+y^2)+ x sin (1/y), y=/0
0, y=0}
question:
1) show that lim (x,y) approaches (0,0) of f(x,y) exists, but neither lim of x-->0 lim y-->0 f(x,y) nor lim of y-->0 lim x-->0 f(x,y) exist.
2) show that lim of x-->0 lim y-->0 g(x,y) exists, but neither lim of y-->0 lim x-->0 g(x,y) nor lim (x,y) approaches (0,0) of g(x,y) exist.
3) show that lim of y-->0 lim x-->0 h(x,y) exists, but neither lim of x-->0 lim y-->0 h(x,y) nor lim (x,y) approaches (0,0) of h(x,y) exist.
Thanks a lot! I need a full explanation! Appreciate it
Explanation / Answer
1).
f(x,y)={x sin (1/y) + y sin(1/x), xy=/0
0, xy=0}
now lim (x,y)--> (0,0) so
when f(x,y) = 0 ;x = 0
lim (x,y)-->0 f(x,y) = 0 , xy = 0
so lim (x,y) exists
when f(x,y)=x sin (1/y) + y sin(1/x); xy=/0
lim(x,y)->(0,0) f(x,y) = 0
now we check for
limx-> 0 [limy->0 (f(x,y))] = limx->0 (x*(fluctuating interval b/w -1 to 1 ))
= some interval between (0,0) = so lim does not exist
limy-> 0 [limx->0 (f(x,y))] = limy->0 (y*(fluctuating interval b/w -1 to 1 ))
= some interval between (0,0) = so lim does not exist
2).
g(x,y)={(xy/x^2+y^2)+ y sin (1/x), x=/0
0, x=0}
when g(x,y)=(xy/x^2+y^2)+ y sin (1/x), x=/0
lim of x-->0 lim y-->0 g(x,y) = limx->0 0 = 0
and when
g(x,y) = 0 ; x = 0
lim of x-->0 lim y-->0 g(x,y) = 0
so limit exists
when g(x,y)=(xy/x^2+y^2)+ y sin (1/x), x=/0
lim of y-->0 lim x-->0 g(x,y) = lim y->0 y*(fluctuating value between -1 to 1 )
== some interval between (0,0)
when
g(x,y) = 0 ; x = 0
lim of y-->0 lim x-->0 g(x,y) = 0
so limit does not exist
3).
h(x,y)= {(xy/x^2+y^2)+ x sin (1/y), y=/0
0, y=0}
when h(x,y)= (xy/x^2+y^2)+ x sin (1/y), y=/0
lim of y-->0 lim x-->0 h(x,y) = lim y->0 0 = 0
when
h(x,y) = 0 , y = 0
lim of y-->0 lim x-->0 h(x,y) = lim y->0 0 = 0
when
h(x,y) = 0 , y = 0
lim of x-->0 lim y-->0 h(x,y) = lim y->0 0 = 0
limit does not exist
when h(x,y)= (xy/x^2+y^2)+ x sin (1/y), y=/0
lim (x,y) -> (0,0) h(x,y) = 0*(fluctuating alue b/w -1 to 1 ) = some value b/w (0,0)
when
h(x,y) = 0 , y = 0
lim (x,y) -> (0,0) h(x,y) = 0
so limit does not exist
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