lim as x,y approaches 0,0 of xysin(1/xy))

Question

lim (x,y) aproaches (0,0) f(x,y)=lim (x,y) aproaches (0,0) (xysim(1/xy))

Answer 1

x y sin (1 / x y)

The x y part will keep getting smaller and smaller in magnitude (closer and closer to 0) as x and y each approach 0.

The function sin ( ) goes back and forth between -1 and 1, never greater than 1 or less than -1.

The sin (1 / x y) part will wiggle back and forth faster and faster as x and y approach 0 because 1 / x y will get larger and larger, but the sin ( ) function itself never goes beyond -1 and 1.

The x y part keeps getting smaller in magnitude while the sin ( ) is stuck between -1 and 1, so the x y part will overpower the sin ( ) part.  The limit should be 0.

You can see a graph by putting this in Google:  plot(x*y*sin(1/(x*y)))

(set the x:  -1 to 1, y: -1 to 1, and z: -2 to 2 and you'll see how it flattens out in the middle, approaching 0)