RECTILINEAR ASYMPTOTES. 291 



268. If -~ increases without limit, and at the same time 



OX 



doc 



xy, has a finite limit, we have an asymptote parallel to 

 ay 



the axis of y. 



Also we may have an asymptote when the limit of 



fi'J* 



x y-.- is infinite, namely in the case where the limit of 



-,- is zero, and the limit of y x-~, which is the intercept 

 dx dx 



on the axis of y, is finite. The asymptote is then parallel to 

 the axis of x. 



269. We will now take some simple examples. 



(1) The equation to the parabola is y'*=lax\ so that 



we have y = 2 Vaic ; therefore j i A /- > hence, when 



((tC y sc 



x increases indefinitely the limit of -,- is zero; but 

 y x-~=(2 ^lax *Jax] = + ^/ax, which has no finite limit. 



Therefore there is no asymptote. 



b* 



(2) The equation to the hyperbola is y*= ^ (a; 2 a 2 ) ; so 



ft 



that we have y=-*J(x*c?}; therefore -> = + -.. 



dx x s -a 3 cf dy , 



and x y- r =x = . Hence the limit of -f- when 



17 dy x x dx 



h fj y* 



x is infinite is + - , and the limit ofx y-r is 0. There- 

 ~ a ^ dy 



fore y = - x is the equation to one asymptote ; and y = x 



a a, 



is the equation to another asymptote. 



3 



(3) Suppose y = - -^ +c to be the equation to a curve, then 

 (xo) 



di/ 2a 3 .,.. dx x k c(x b} 3 



we have -r- = -, TTT> and xy'-j- =x + h - , . 



dx (x-b) 3> y dy 2 2a 3 



