( 200 ) 



CHAPTER XIX. 



ASYMPTOTES. 



2G6. SUPPOSE one or more of the branches of a curve to 

 extend to an infinite distance from the origin, and that at 

 successive points of such a branch we draw tangents. Then 

 two different cases may exist with respect to the directions of 

 these tangents ; they either, as we pass from point to point 

 along the curve, approach some definite limit or they do not. 

 And with respect to the position of these tangents, two cases 

 are possible ; the intercepts cut from the axes of co-ordinates 

 either tend to a finite limit or they do not. If the direction 

 has a limit, and one or both of the intercepts a limit, there 

 exists a straight line towards which the successive tangents 

 continually approach. Such a straight line is called an 

 asymptote to the curve ; hence we have the definition which 

 follows. 



267. DEFINITION. An asymptote to a curve is the limit- 

 ing position of the tangent when the point of contact moves 

 to an infinite distance from the origin. 



To find whether a proposed curve has an asymptote, we 



must first ascertain if -j- has a limiting value as we proceed 

 to an infinite distance from the origin. If it has not there ia 

 generally no asymptote. If -J- has a limiting value, we must 

 then ascertain if the intercept on the axis of x, which by 



ft fj* 



Art. 260 is xy-j-, has a limiting value. Suppose it has, 



and let it be denoted by c while JM denotes the limit of -j- , 

 then y p^x c) is the equation to an asymptote. 



