190 On Taylor's Theorem. 



&c. approximate to the limit (a); «, a', a", &c. to the limit («); 

 and so on : and if 



_§_ 

 2 



s 



then — must lie between the greatest and least of the fol- 



Mi 



lowing fractions, if every term of all the preceding denomina- 

 tors be positive : 



£! W & &c 



(«)' ®' (r)' 



. Theorem 3. If p and y be two quantities which have the 

 limit a 9 then the limits of 



p-q h 



must be the same, in whatever manner p and q are supposed 

 to vary, provided the limits be severally equal to «. 



Theorem 4-. If $ x be a function which does not become 

 infinite for any value of x between a and b, then 



$(x + h) — $x 



must have a finite limit, when h is diminished without limit, 

 for some values of x lying between a and b. 



Theorem 5. If any value of x be assigned, the differential 

 coefficient of any function of x is finite, either for that value 

 of x, or for one within any given degree of nearness to it. 



Theorem 6. If the differential coefficient of <$> x be finite 



for every value of x between a and b, then — — -5 — must lie 



between the greatest and least value of the differential coeffi- 

 cients between <p' a and <p ! b. 



In what follows, I shall use the abbreviation a + Qh for a 

 quantity lying between a and a-\-h, both inclusive; that is, 

 lies between and 1, both inclusive. 



Theorem 7. If $' {a + Qh) be always positive, $(a + h) is 



