Theory of Limits. 141 



cumference of the circle, or in other words the perimeter of the 

 polygon can never equal the circumference of the circle, but as 

 the process of doubling the number of sides is continued, the 

 perimeter of the polygon differs less and less from the circumfer- 

 ence of the circle, hence the circumference of the circle is the 

 limit of the perimeter of the inscribed polygon. 



5. The student should not infer from what has been said that 

 all variables have limits! In fact, the truth is quite the contrary, 

 for most variables do not have limits. Thus, in the illustration of 

 the moving point given above, the variable does not have a limit 

 if we suppose the point to move at a uniform rate. For, if the 

 velocity is uniform, it is a mere question of time until the moving 

 point passes /?, or, in fact, any other point to the right of B, how- 

 ever remote. Much more would this be true if the point moved 

 with increasing instead of uniform velocity. 



Again, consider the fraction 



X 



If X be supposed to change in value, the value of the fraction 

 changes and is itself a variable. Now suppose x to decrease in 

 value. It is plain t\idX the value of the fraction i?icreases without 

 limit as x decreases. In other words, the value of the fraction can 

 be made as large as we please by taking x small enough. Hence, 

 as X decreases, the value of the fractio7i has no limit. 



6. It follows immediately from the definition of a variable, 

 that the difference betzvee7i a variable and its limit is a variable 

 ivhose limit is zero. 



For if X be a variable whose limit is a, then x may be made to 

 differ from a by as small a quantity as we please, hence a—x may 

 be made as small as we please; yet as x can never equal a, a—x 

 can never equal zero, hence a—x is a variable, whose limit is zero. 



7. Theorem. //' taw variables are always equal and each 

 approaches a limit, the limits must be equal. 



Let X and V be the variables, and let limit x=a and limit j'=^. 



