INDETERMINATE EXPONENTIAL FORMS. 133 



Again, x m (log x) n , 



where m and n are positive, takes the form x oo , when x 



Here < takes the form - 



when x = ; its limit is the same as that of 



x (log ic)' m 

 which does not assist us. 



If we assume x = 6*", then x m (log x}* becomes 



the value of this, when y is oo , is 0. See Art. 153. 



The result in this case should be carefully noticed, as it is 

 frequently wanted in mathematical investigations. 



159. Forms 0, 00, 1". 



Let <> (a?) and ty (x) be two functions of x, such that when 

 x = a, the expression 



assumes one of the forms 0, oo , I 08 ; it is required to find the 

 limiting value of this expression. 



Since < (x) = 



we have 



Now i/r (a?) log </> (#) in each of the proposed cases takes 

 the form x oo , and its limiting value can be found by 

 Art. 158, and thus the value of {< (x}}^ x) becomes known. 



For example, x x , when x = 0, takes the form ; 



and a? logic = 0, when x = 0, (Art. 158) ; 

 therefore, of = 1, when x = 0. 



