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I. The definition of a limit. 2. Problems, by which limits fo defined are obtained. 

 3. Properties of thofe] limits, 4. Application of thofe properties to geometrical demon- 

 ftrations, to algebraical procefles, and {o the tranflation ofphyfical relations into mathe- 

 matical ones. 



1. Definition A limit of a variable quantity, whether a Cmple magnitude or the 

 magnitude of a ratio is that quantity to which the variable quantity may fo approach 

 as to differ from it in magnitude by lefs than .any afiigned or ftated quantity. 



J. A variable fecant drawn from a given point without a circle, may be drawn, 

 fo as to differ from the tangent in magnitude, lefs than by any magnitude that can be 

 afligned. For, affign a difference, then the poCtion of the fecant is determined, and 

 therefore between the fecant and tangent another fecant can be drawn. The magnitude 

 of the tangent is a limit of the magnitude of the fecant. But the fecant is not faid to 

 become the tangent. 



2. A limit of the quantity xj—Oi where s is greater than a is 2a for "^ — °' may 



X — a * — a 



be made to differ from 2a by a lefs magnitude than any afligned one. But za cannot 



with propriety be faid to be one of the values of ^•'--''- or to be equal to £i_ 



X — a 0. 



3. The limiting ratio of the increment of the abfciffa of a curve to the increment 

 of the ordinate is equal to the ratio of the fubtangent to the ordinate. The ratio of 

 the increments themfelves is never equal to the ratio of the fubtangent to the ordinate. 



II. Prob. It is therefore a mathematical problem to find a limit fo defined, of a 

 variable quantity, when it admits of one. The folution of the problem is evidently had 

 if the variable quantity or its equal can be expreffed by the fum or difference of 

 two quantities one fixed and the other variable, the variable one admitting a lefs value 

 than any afligned one. The fixed quantity is a limit. 



I. Suppofing X any magnitude greater than a, and a limit of '' ^ — "'^ be required. 



x—a 

 Becaufe "'"''' = x+a = la+e (putting x = a-\-e). Therefore as e may be 



x—a 

 lefs than any afligned magnitude 2a is the limit required. 



To determine the limiting ratio of increment of the abfcifla to the increment of the ordinate. 

 The incr. of the abfciffa : increm. ordinate : : fubtangent -J- 11 : ordinate, by fimilar triangles. 

 As the ordinates may approach fo that v may become lefs than any afligned quan- 

 tity, the limit of the latter, and therefore of the former ratio is the ratio of the 

 fubtangent to the ordinate. This method is frequently applicable when the terms of 

 the propofed ratio are both variable. A ratio is found equal to the given ratio, one 

 of the terms of which is fixed and the other variable. A limit of the variable term 

 is then found, and thence the limit of the propofed ratio is had. 



( U 2 ) In 



