150 Algebra. 



^. , limit ( mx ) 



2. Find ^ \— ^ Y 



X ^ oKpx —ax) 



^. , limit (a- — x" 



?. Find ■<— 



X ^ a I a — X 



^. , limit f rr4-//;-^--^-M 



4. Find , ^ -) , \ 



^ h ^ o { h j 



limit (.r^+i ) 



5-. Find ^ V ^ i 



X ^ \ I X-— I ) 



^^ limit \ x"—a" ) 



^. Prove ^ \ \. =na" \ 



X ^ a I x—a I 



23. Limit of the Sum of a Decreasing Geometrical 

 Progression as n Increases. It was noticed in Chapter VIII 

 that if the ratio of a geometrical progression is less than unity, 

 each term of the series is necessarily less than the one preceding 

 it. In this case the series is called a dea-easing progression. 



In the case of a decreasing geometrical progression, it is a little 

 better to write the expression for the sum of the series in the form: 



I — r 

 Now if we like we may consider ;/ a variable, and then the 

 two sides of this equation are two variables that are always 

 equal. Therefore, their limits are equal. Whence we may write 



lim s as ;/ mcreases=lim^ ^ - — }as n increases. 



I I — r J 



Now since /is less than i, the term r" continually approaches 

 the limit o as )i increases. Whence taking the limit of the right 

 hand member of the equation, we may write : 



.. . a 



lim .v as n increases= 



I — /'. 



24. Examples. 



/. Find the limit of the progression r-SSSS + y* ^^ ^^ increases. 

 Here ^=Y\ and /'=yV. 



Whence, lim s= ' ^ ^ = 4. 



^~T0 



Therefore, lim .3333+ = rv- 



