INDETERMINATE FORMS 
213 
If a 1, write 
(log x) a _ Tlog x] a 
xP /a 
Since the variable standing within the brackets approaches 0, and 
since a is positive, the whole right-hand side approaches 0, and thus 
the original variable approaches 0 in all cases.* 
3. The Limit 0-ao. If f(x) approaches 0 and <j>(x) becomes in 
finite, the limit approached by the product may often be determined 
by writing 
= №) 
or 
thus throwing the variable into the form discussed in § 1 or § 2. 
Example, lim x log x. 
a; log a; -= lim(a;logr:B) = lim —^ = lim(— x)= 0. 
When one factor in this last exponent approaches 0 and the other 
becomes infinite, the limit of the exponent is of the type considered 
in § 3. Thus we are led to the limits which may be symbolized as 
Example, lim x x . 
Since lim (a; log a;) = 0 by § 3, 
lim x x = lim e xlogx = 1. 
The limit of f(x) — <f> (x), where f(x) and both become infinite 
with the same sign, is usually best treated by special methods. 
Example, lim ^ Va; 2 + 1 — x\. 
\ VX 2 -+ 1 — x\\^\/x 2 + 1 x\ _ 1 
Write Va: 2 + 1 — x = 
VaJ 2 + l + ic Va; 2 -f 1 + x 
* A thorough appreciation of the meaning of the graph of the function y = x n , 
Introduction to the Calculus, p. 160, is important in the study of the present 
chapter.