Multivariate Calculus - Bruce E. Shapiro

LECTURE 10. LIMITS AND CONTINUITY 71
Example 10.2 Show that
does not exist.
x 4 − y 4
lim
(x,y)→(0,0) x 4 + y 4
Solution. We calculate the limit in two directions: along the x-axis and along the
y-axis.
Along the x-axis, y=0, so that
Along the y-axis, x=0, so that
x 4 − y 4
lim
(x,y)→(0,0),x−axis x 4 + y 4 = lim x 4 − 0 4
x→0 x 4 + 0 4 = 1
x 4 − y 4
lim
(x,y)→(0,0),y−axis x 4 + y 4 = lim 0 4 − y 4
y→0 0 4 + y 4 = −1
Since the limits along the different paths are unequal we may conclude that the limit
does not exist.
Definition 10.4 A function is said to be continuous at a point (a, b) if
lim f(x, y) = f(a, b)
(x,y)→(a,b)
A function is said to be continuous on a set S if it is continuous at every point
in S.
Theorem 10.1 The following classes of functions are continuous:
(a) Lines
(b) Polynomials
(c) Composite functions of continuous functions, e.g., f(g(x, y))
(d) Rational functions except where the denominator equals zero.
Example 10.3 Determine where the function
is continuous.
f(x, y) = x4 − y 4
x 4 + y 4
Solution. f(x, y) is a rational function. Therefore it is continuous except where the
denominator is zero. The function is not continuous when
x 4 + y 4 = 0
which only occurs at the origin. Therefore the function is continuous every except
for the origin.
Math 250, Fall 2006 Revised December 6, 2006.