292 RECTILINEAR ASYMPTOTES. 



As x approaches b, y and increase without limit. The 



dx 



limit of x y -,- is b, and, by Art. 268, there is an asymp- 

 ay 



tote parallel to the axis of y, having for its equation x = b. 



270. An asymptote may also be defined as a straight line, 

 the distance of which from a point in a curve diminishes with- 

 out limit as the point in the curve moves to an infinite distance 

 from the origin. 



Suppose y = px + ft 



the equation to a straight line, and 

 y = px + ft + v 



the equation to a curve, then if v diminish without limit as 

 x and y increase without limit, the straight line will be an 

 asymptote to the curve. For if x, y, be the co-ordinates of 

 a point in the curve, the perpendicular distance of that point 

 from the straight line is 



y fix ft 



' 



ir i * 



and this diminishes without limit when x and y increase 

 without limit. 



271. That the two definitions of an asymptote lead in 

 general to the same results may be seen by considering differ- 

 ent examples, or by the following proof. Let y = ^x + ft + v 

 be the equation to a curve, where /A and ft are constants, and 

 v diminishes without limit as x and y increase without limit. 

 From the given equation 



X 



Hence u, is the limit of " when x and y increase without 



a; 



limit. But, by Art. 148, 



dy 



the limit of - = the limit of or -f- . 

 x ax 



