CHAPTER XI. 



THEORY OF LIMITS. 



1. Definition. When a quantity preserves its value un- 

 changed in the same discussion it is called a Constant, but when 

 under the conditions of the problem a quantity may assume an 

 indefinite number of values it is called a Variable. 



Constants are usually represented by the first or intermediate 

 letters of the alphabet and variables by the last letters. 



The notation by which we distinguish between constants and 

 variables is the same as that by which we distinguish between 

 known and unknown quantities, but it must not be thought that 

 any analogy is intended to be pointed out by this fact. When we 

 are discussing a problem in which both constants and variables 

 appear we usually do not care whether the constants are known 

 or unknown. 



2. Definition. When a variable in passing from one value to 

 another passes through all intermediate values it is called a 

 conthiuous variable; when it doe's not pass through all intermed- 

 iate values it is called a discontinuous variable. 



3. Definition. When a variable vSo changes in value as to 

 approach nearer and nearer some constant quantity which it can 

 never equal, 3^et from which it may be made to differ by an 

 amount as small as we please, this constant is called the Limit of 

 the variable. 



4. Illustrations. If a point move along a line AB, starting 



A n 



■I ■ II 



at A and moving in such a way that the first second the point 

 moves one-half the distance from A to />*, the second second one- 

 half the remaining distance, the third second one-half the distance 

 which still remains, and so on ; then the distance from A to the 

 moving point is a variable whose limit is the distance AB. For, 

 no matter how long the point has been moving, there is vStill some 



