294 CURVILINEAR ASYMPTOTES. 



,-, a + sin x 



Also y [J>x = f-\ . 



x 



fit i r*r\& if* 



And 



dy _ . cos x a + sin x 

 dx x x* 



dy ~ 2 (a + sin x) 



therefore y -, - x = B cosaH . 



dx x 



Here we cannot assert that y fix and y --JL x have the 



same limit : the limit of the former is B, but the latter cannot 

 be said to have a limit, on account of the term cos x, which 

 does not tend to any limit as x increases indefinitely. In 

 this case the curve 



_ a + sin x 

 y=Ax+B+ 



x 



has an asymptote according to the definition of Art. 270, 

 namely, y = Ax + B, but not according to the definition of 

 Art. 267. 



The demonstration in Art. 270 might, of course, start 

 with the equation x = py + /? + v ; so that, should the asymp- 

 tote be parallel to the axis of y, by taking the second form 

 we avoid having p, infinite. 



273. We have hitherto confined ourselves to rectilinear 

 asymptotes ; we now extend the definition to curvilinear 

 asymptotes. 



DEFINITION. When the difference of the ordinates of two 

 curves corresponding to a common abscissa diminishes without 

 limit, or the difference of the abscissas corresponding to a 

 common ordinate diminishes without limit, as we pass from 

 point to point along either curve, each curve is said to be an 

 asymptote to the other. 



Hence, if the equation to a curve can be put in the form 



BI B a B 9 



11 w ^ <T^ * * ** 



.' */ **/ 



then y = A x n + Ap*' 1 + ...+ A^x + A n 



