Page 136 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 6] PARTIAL DERIVATIVES 127

x 2 z 2
Trace on xz plane ðy ¼ 0Þ is the hyperbola ¼ 1, y ¼ 0.
a 2 c 2
Trace on any plane z ¼ p parallel to the xy plane is the ellipse
x 2 y 2
2 2 2 þ 2 2 2 ¼ 1
a ð1 þ p =c Þ b ð1 þ p =c Þ
As j pj increases from zero, the elliptic cross section increases in size.
The surface is a hyperboloid of one sheet (see Fig. 6-8).

LIMITS AND CONTINUITY
2
6.4. Prove that lim ðx þ 2yÞ¼ 5.
x!1
y!2 Fig. 6-8
Method 1,using definition of limit.
2
We must show that given any > 0, we can find > 0suchthat jx þ 2y 5j < when 0 < jx 1j < ,
0 < j y 2j < .
If 0 < jx 1j < and 0 < j y 2j < ,then1 < x < 1 þ and 2 < y < 2 þ ,excluding
x ¼ 1; y ¼ 2.
2
2
2
Thus, 1 2 þ < x < 1 þ 2 þ and 4 2 < 2y < 4 þ 2 . Adding,
2
2
2
2
5 4 þ < x þ 2y < 5 þ 4 þ 2 or 4 þ < x þ 2y 5 < 4 þ 2
2
2
Now if @ 1, it certainly follows that 5 < x þ 2y 5 < 5 , i.e., jx þ 2y 5j < 5 whenever
0 < jx 1j < ,0 < j y 2j < . Then choosing 5 ¼ , i.e., ¼ =5(or ¼ 1, whichever is smaller), it
2
2
follows that jx þ 2y 5j < when 0 < jx 1j < ,0 < j y 2j < , i.e., lim ðx þ 2yÞ¼ 5.
x!1
y!2
Method 2,using theorems on limits.
2
2
lim ðx þ 2yÞ¼ lim x þ lim 2y ¼ 1 þ 4 ¼ 5
x!1 x!1 x!1
y!2 y!2 y!2
2
6.5. Prove that f ðx; yÞ¼ x þ 2y is continuous at ð1; 2Þ.
2
By Problem 6.4, lim f ðx; yÞ¼ 5. Also, f ð1; 2Þ¼ 1 þ 2ð2Þ¼ 5.
x!1
y!2
Then lim f ðx; yÞ¼ f ð1; 2Þ and the function is continuous at ð1; 2Þ.
x!1
y!2
Alternatively, we can show, in much the same manner as in the first method of Problem 6.4, that given
any > 0wecan find > 0suchthat j f ðx; yÞ f ð1; 2Þj < when jx 1j < ; j y 2j < .
2
x þ 2y;
ðx; yÞ 6¼ð1; 2Þ .
0; ðx; yÞ¼ ð1; 2Þ
6.6. Determine whether f ðx; yÞ¼

(a) has a limit as x ! 1and y ! 2, (b)is continuous at ð1; 2Þ.
(a)By Problem 6.4, it follows that lim f ðx; yÞ¼ 5, since the limit has nothing to do with the value at ð1; 2Þ.
x!1
y!2
(b)Since lim f ðx; yÞ¼ 5 and f ð1; 2Þ¼ 0, it follows that lim f ðx; yÞ 6¼ f ð1; 2Þ. Hence, the function is
x!1 x!1
y!2 y!2
discontinuous at ð1; 2Þ:
8 2 2
x y
<
2
6.7. Investigate the continuity of f ðx; yÞ¼ x þ y 2 ðx; yÞ 6¼ð0; 0Þ at ð0; 0Þ.
:
0 ðx; yÞ¼ ð0; 0Þ
Let x ! 0 and y ! 0insucha way that y ¼ mx (a line in the xy plane). Then along this line,
2
2 2
2
x y 2 x m x 2 2 1 m 2
lim ¼ lim ¼ lim x ð1 m Þ
2
2 2
2
x!0 x þ y 2 x!0 x þ m x 2 2 ¼ 1 þ m 2
y!0 x!0 x ð1 þ m Þ