2 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Good examples of limits are and -- , when x 



x x 



tends to 0, for we know that x being the circular measure 



. sina? x 2 . sin a? 



ot an angle < 1 and > 1 , whence 1 > and 



x 4 ' x 



2 * 



< , whence the difference 1 continually diminishes 



4 7 x 



as x diminishes, and may be made less than any assignable 

 quantity, however small, before a? = 0; whence the limit of 



sina? . ,.. -11 since . 



is 1. When x is absolute zero, is unmeaning. 



x x 



/m 11 ir\ 

 This result is however, for shortness, written ( - - J = 1 ; 



but the true meaning of such equations should always be 

 borne in mind. 



tana; x sin a; .. . 



ho also = - : therefore, the limit ot 



x cos a; x 



tana? ....,, sin su . . , , ,, 



= the limit of - = 1, which, as before, is written 

 x x 



C4"flTl f y*\ 

 ) = 1. These results are best illustrated by 

 X ' X=0 



drawing the curves y = si no:, ?/ = tan#; i.e. measure off on 

 the straight line Ox any length OM (fig. 1) containing as 

 many units of length as x contains of angle (in circular 

 measure), and then draw I/Pat right angles to Ox and re- 

 presenting on the same scale sin a?, then as x increases from 

 to ^TT, y increases from to 1 and is positive ; from X = \TT 

 to TT, y decreases again from 1 to 0, and is positive, while 

 from TT to 2-7T, the same arithmetical values recur in the same 

 order, but sin a? is negative, and the corresponding curve 



is on the negative side of the axis of x. Here - is 



always represented by tanPOJ/, and since the limit has 

 been shewn to be 1 , the limiting value of the angle POM, 

 when M moves up to A, is ^TT; but the limiting position 

 of PO when J/, and therefore P moves up to (9, is what 

 we mean by the tangent line to the curve at 0. Hence 



