132 INDETERMINATE PRODUCT. 



e* 1 



, or - x e x is of course infinite when x = 0. 



x x 



e 

 Hence, is or oo when x approaches the limit 0, 



flp 



according as we suppose x negative or positive. 



158. Form x co . 



Suppose <f>(x) and -^r(x} two functions of x, such that 

 (j>(a) = 0, and ^(a) = ao ; it is required to find the limit of 

 <j> (x) A/T (x) as x approaches a. 



and as the fraction on the right-hand side takes the form 

 - when x = a, its limiting value may be found by rules 

 already given. 



/ SC\ TTOC 



For example, let <j> (x) = log f 2 j , and ty (x) = tan . 



Here $ (x) i/r (x) takes the form x oo when x = a. 



log (2--) 

 TTX \ a) 



-LllLil - ^ | j ww ~. 



2a 



cot 

 2a 



The limit of this when x = a, is found by making x = a in 

 1 1 



2a* . ., 

 sm 



, . , . . 2 



which gives 



7T 



