90] 
MISCELLANEOUS EXAMPLES ON CHAPTER V 
183 
Another method of stating the definition is this: $ (x, y) is continuous 
for x — y = t] if f if, rj) is defined and $ (x, y)-*~4> (& v) when x-*~£, y-*~n in 
any manner. This statement is apparently simpler; but it contains phrases 
the precise meaning of which has not yet been explained and can only be 
explained by the help of inequalities like those which occur in our original 
statement. 
It is easy to prove that the sums, the products, and in general the 
quotients of continuous functions of two variables are themselves continuous, 
A polynomial in two variables is continuous for all values of the variables; 
and the ordinary functions of x and y which occur in every-day analysis are 
generally continuous—i.e. are continuous except for pairs of values of x and 
y connected by special relations. 
MISCELLANEOUS EXAMPLES ON CHAPTER Y. 
1. Show that, if neither a nor b is zero, ax n + hx n ~ 1 + ...+& = ax n (1 + 6*), 
where e x is of the first order of smallness when x is large. 
2. If P (x) = ax n + bx n ~ 1 +... + k, then as x increases P (x) has ultimately 
the sign of a; and so has P(x + A) — P(x), if A is any constant. 
3. Show that in general 
(ax n + hx n " 1 +... + k)j(A x n + Bx n ~ 1 + ... + K) = a + (ßjx) (1 + 6*) 
where a=a/A, ß=(bA - aB)jA 2 and e x is of the first order of smallness when 
x is large. Indicate any exceptional cases. 
4. Express (ax 2 + bx + c)/(A x 2 + Bx + 0) 
in the form a + (ßjx) + (yjx 2 )(\ + e x ), 
where e x is of the first order of smallness when x is large. 
5. Show that lim fix (x + a) - f x] = 
[Use the formula *J(x+a)- Jx = a/{f(x+a) + IK , x}.] 
6. Show that f(x + a) = s/x + % (ajfx) (1 + e x ), where e x is of the first order 
of smallness when x is large. 
7. Find values of a and /3 such that v /(ax 2 + 2bx + c) — ax — ß has the limit 
0 as x-*~x> ; and prove that lim x{J(a:v 2 +2bx+c) — ax — ß} = (ac-b 2 )j2a. 
8. Evaluate lim ¿’{v/^ + v/^ + l)]-#,^}. 
9. If ax 2 + 2bxy + cy 2 + 2dx+2ey=0 and A = 2bde - ae 2 - cd 2 , then one 
value of y is given by y = ax + ßx 2 + yx 3 (1 + e x ), where 
a= - d/e, ß = A/2e 3 , y = (cd-be) A/2« 5 , 
and ( x is of the first order of smallness when x is small. 
[If y-ax—rj we have 
— 2erj = ax 2 + 2bx + ax) + c (rj + aX) 2 = Ax 2 + 2Bxrj + Crj 2 ,