2. LIMITS AND CONTINUITY
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sin(x2 + y 2)
Example. Find lim
.
x2 + y 2
(x,y)!(0,0)
We use polar coordinates to find the indicated limit, if it exists. Note that
(x, y) ! (0, 0) is equivalent to r ! 0. We have
sin(x2 + y 2)
sin(r2)
lim
= lim
= (by l’Hopitals”s rule)
r!0
x2 + y 2
r2
(x,y)!(0,0)
2r cos(r2)
lim
= lim cos(r2) = cos 0 = 1.
r!0
r!0
2r
sin(x2 + y 2)
Thus lim
= 1.
x2 + y 2
(x,y)!(0,0)
Continuity
Definition. Suppose f (x, y) is defined in the interior of a circle centered
at (a, b). We say f is continuous at (a, b) if
lim
(x,y)!(a,b)
f (x, y) = f (a, b).
If f (x, y) is not continuous at (a, b), then we call (a, b) a discontinuity of f .
Definition.
(1) An open disk is the interior of a circle, i.e., all points inside but not on the
circle.