ll' INVESTIGATION OF A LIMIT. 



the different portions of the subject he is studying, and of 

 selecting the definitions necessary to be understood ; and in 

 reading a work on the Differential Calculus, he must be 

 satisfied at first with reflecting upon the meaning of the 

 definitions, and examining whether the deductions drawn by 

 the writer from those definitions are correct. There are 

 innumerable applications of the elementary principles of the 

 Differential Calculus, as will be seen in the Chapter on 

 Expansions and those following it, but we shall at first 

 confine ourselves merely to the logical exercise of tracing the 

 consequences of certain definitions. 



A difficulty of a more serious kind which is connected with 

 the notion of a limit, appears to embarrass many students 

 of this subject, namely, a suspicion that the methods em- 

 ployed are only approximative, and therefore a doubt as to 

 whether the results are absolutely true. This objection is 

 certainly very natural, but at the same time by no means 

 easy to meet, on account of the inability of the reader to 

 point out any definite place at which his uncertainty com- 

 mences. In such a case all he can do is, to fix his attention 

 very carefully on some part of the subject, as the theory 

 of expansions for example, where specific important formulae 

 are obtained. He must examine the demonstrations, and if 

 he can find no flaw in them, he must allow that results 

 absolutely true and free from all approximation can be le- 

 gitimately derived by the doctrine of Limits. 



23. The demonstration in Arts. 15, 16 of the proposition 



/ IN* 



that ( 1 + - ] tends to some fixed limit as x increases in- 



V 7 



definitely, has been given in several elementary works on 

 the Differential Calculus, and it is accordingly retained here. 

 But the following method, in which the Binomial Theorem 

 is not assumed, is worthy of notice. 



"We shall first establish two inequalities. 



If /3 and X are positive quantities, and \ greater than 

 unity, 



is greater than l + X/3 (I). 



