DIFFERENTIAL AND INTEGEAL 

 CALCULUS. 



THE first thing necessary in beginning the study of 

 the Differential Calculus, is to have a clear idea of what 

 is meant by the limit of a varying quantity, which, under 



oo 

 certain circumstances, assumes the unmeaning form -or . 



oo 



The definition of a limit in such a case is as follows: if 

 U be a varying quantity, which on a certain hypothesis 

 assumes an unmeaning form, and A the limit of this 

 quantity, then as U is approaching its ultimate form, the 

 difference between Z7and A continually diminishes, and may 

 be made less than any assignable quantity however small, 



before U assumes the form - ; but this difference does 



not become absolute zero before that point. 



Of course we might say that, e.g., when x approaches 

 the value 2, x 2 approaches the value 4, and thus that 4 

 is the limit of x z when x approaches 2, but in such a case, 

 where the function (x*) has always an intelligible meaning, 

 it is simpler to say that x 2 is 4 when x = 2 ; but suppose 



1 41 



we have x = 2 + - t , and therefore x 2 = 4 -f 3 -f , then 



as z is indefinitely increased x continually approaches 2, 

 and x* continually approaches 4, and the differences x 2, 

 a? 2 4 may be made less than any assignable quantity 

 however small ; and in such a case we should say that x 

 has 2, and x* has 4, for its limit, since in this case x is 

 not absolutely = 2, nor x* = 4, for any finite value of z 

 however large. 



