Page 127 - Schaum's Outline of Theory and Problems of Advanced Calculus

118 PARTIAL DERIVATIVES [CHAP. 6

Fig. 6-3

LIMITS
Let f ðx; yÞ be defined in a deleted neighborhood of ðx 0 ; y 0 Þ [i.e.; f ðx; yÞ may be undefined at
ðx 0 ; y 0 Þ]. We say that l is the limit of f ðx; yÞ as x approaches x 0 and y approaches y 0 [or ðx; yÞ
approaches ðx 0 ; y 0 Þ] and write lim f ðx; yÞ¼ l [or lim f ðx; yÞ¼ l]if for any positive number we
x!x 0
ðx;yÞ!ðx 0 ;y 0 Þ
y!y 0
can find some positive number [depending on and ðx 0 ; y 0 Þ,in general] such that j f ðx; yÞ lj <
whenever 0 < jx x 0 j < and 0 < j y y 0 j < .
2
2
If desired we can use the deleted circular neighborhood open ball 0 < ðx x 0 Þ þð y y 0 Þ < 2
instead of the deleted rectangular neighborhood.

3xy
EXAMPLE. Let f ðx; yÞ¼ if ðx; yÞ 6¼ð1; 2Þ .As x ! 1 and y ! 2[or ðx; yÞ!ð1; 2Þ], f ðx; yÞ gets closer to
0 if ðx; yÞ¼ ð1; 2Þ
3ð1Þð2Þ¼ 6 and we suspect that lim f ðx; yÞ¼ 6. To prove this we must show that the above definition of limit with
x!1
y!2
l ¼ 6issatisfied. Such a proof can be supplied by a method similar to that of Problem 6.4.
Note that lim f ðx; yÞ 6¼ f ð1; 2Þ since f ð1; 2Þ¼ 0. The limit would in fact be 6 even if f ðx; yÞ were not defined at
x!1
y!2
ð1; 2Þ. Thus the existence of the limit of f ðx; yÞ as ðx; yÞ! ðx 0 ; y 0 Þ is in no way dependent on the existence of a value
of f ðx; yÞ at ðx 0 ; y 0 Þ.
Note that in order for lim f ðx; yÞ to exist, it must have the same value regardless of the
ðx;yÞ!ðx 0 ;y 0 Þ
approach of ðx; yÞ to ðx 0 ; y 0 Þ. It follows that if two different approaches give different values, the
limit cannot exist (see Problem 6.7). This implies, as in the case of functions of one variable, that if a
limit exists it is unique.
The concept of one-sided limits for functions of one variable is easily extended to functions of more
than one variable.

1
1
EXAMPLE 1. lim tan ð y=xÞ¼ =2, lim tan ð y=xÞ¼ =2.
x!0þ x!0
y!1 y!1
1
EXAMPLE 2. lim tan ð y=xÞ does not exist, as is clear from the fact that the two different approaches of Example
x!0
y!1
1give different results.
In general the theorems on limits, concepts of infinity, etc., for functions of one variable (see Page
21) apply as well, with appropriate modifications, to functions of two or more variables.