RECTILINEAR ASYMPTOTES. 293 



Also /3 is the limit of y \LX ; but /j, = the limit of ^- ; 



therefore in general /3 = the limit of y -' r- x. Hence the 



equation to the tangent to the curve at the point (x, y), 

 which is 



becomes, when x and y are indefinitely increased, 



that is, the equation to the asymptote found according to the 

 first definition is the same as the equation found according to 

 the second definition. 



272. We say in the last Article that in general the limit 

 y p,x = the limit of y 

 the equation to a curve is 



of y p,x = the limit of y -jf x. Suppose, for example, that 



therefore - = A H - + ;, 



x xx 



Hence a = the limit of - = A, and 



T* 





w-iiaj5+-. 



x 



A i G/'lf i Cv 



Also ~ = A 5- 



/y /y <>* 



vvt^/ *O 



therefore w ~ x = B-\ . 



cfx a? 



Here y -j-x and y px have the same limit, namely 5. 



j. _j a + sin 05 

 But suppose y = Ax + B-\ . 



Here, as before, p = A. 



