EXAMPLES OF LIMITS. 11 



a v _ i 



Therefore the limit of - when v = 



21. The following results will be found in works on 

 Trigonometry. If the variable x diminish indefinitely 



, ,. . ,, tan x 

 the limit of -- = 1, 

 x 



. ,...', sin" 1 a; 

 the limit of - = 1, 

 x 



, ,. . , tan~'a; 

 the limit of - = 1. 



22. A few general remarks may be made at the close 

 of this Introductory Chapter. It frequently happens that 

 a person commencing this subject is discouraged at the outset 

 because he cannot discover or imagine any practical appli- 

 cation of the somewhat abstruse points to which his attention 

 is directed. From what he remembers of the early portions 

 of those branches of mathematics with which he is already 

 acquainted, he is led to expect that almost as soon as he 

 begins the Differential Calculus, he will be able to compre- 

 hend its general scope, and to make use of it in solving 

 algebraical and geometrical examples ; and being disap- 

 pointed in this expectation, he is apt to imagine as a reason 

 for it, that he has not correctly understood the elementary 

 principles of the subject. It may, therefore, be of some 

 service to assure him, that the difficulty of which he com- 

 plains is probably owing much more to the nature of the 

 subject than to his own want of comprehension. The student 

 must, of course, leave to his teacher the task of arranging 



