4 EXAMPLES OF A LIMIT. 



make a; =100,000, and the required result is obtained. It 

 we wish y to differ from unity by a quantity less than 



, make x = 1,000,000, and the required result is 

 1,000,000 



obtained. Under these circumstances we say "the limit of 

 y when x increases indefinitely is unity." 



8. The importance of the notion of a limit cannot be 

 over-estimated ; in fact the whole of the Differential Calculus 

 consists in tracing the consequences which follow from that 

 notion. The student has probably already fallen upon cases 

 in which the word limit has been used, to which it will be 

 useful to recur. For example, the sum of the geometrical pro- 

 gression 1+^ + 1 + ^+ continued to n terms is 2 ^ , 



O 



and hence he has deduced the result that the limit of the 

 series when the number of terms is indefinitely increased 

 is 2. 



9. A very important example of a limit occurs in works 

 on Trigonometry. It is there shewn that if 6 denote the 



circular measure of an angle, the fraction ^ will, if 9 be 

 diminished indefinitely, approach as near as we please to 

 unity. In other words the limit of - , as d continually 

 diminishes, is unity. We shall express this by saying "the 



limit of - , when = 0, is unity;" that is, we use the 

 u 



words " when = 0" as an abbreviation for " when is 

 continually diminished towards zero," or for "when 6 is 

 diminished without limit'' 



10. The proposition "the limit of~- , when 0=0, is unity" 



s\ 



is sometimes expressed thus, " = 1, when 0=0," or 



u 



" sin = 0, when = 0." It must however be most carefully 

 remembered that such expressions are only abbreviations and 

 cannot be understood absolutely. In like manner the result 



