142 INDETERMINATE FORMS. 



Thus we obtain again the form - , and we may continue in 



the ordinary way the process of evaluation. We may how- 

 ever obtain the result more easily by arranging the fraction 

 we have now to evaluate thus : 



2(1+ 2# 2 ) cos* x x 



(1 + 3? + x 4 ) (1 + cos 2 x) sin x ' . 

 Here the first factor is not indeterminate when x = ; its 

 value is then unity. The second factor takes the form - , 



and its limiting value is known to be unity. Thus unity is 

 the required limiting value of the original expression. 



Or the original expression may be evaluated in the follow- 

 ing manner. It may be put in the form 



cos x log (1 + x* + x*} 

 sin* x 



Now cos x = 1 when x ; we need not then pay any atten- 

 tion to this factor, but consider that we have to evaluate 



log (1 + a? + x*} 

 sin* x 



when x = ; and we may proceed in the usual way to dif- 

 ferentiate the numerator and denominator. Or if we are 

 allowed to use the results of the expansions of functions we 

 have 



x*-% (x* + x*}* + 



= 1 when x = 0. 



