64 



in which we might suppose a the least exponent ; and then, while x 

 approached to 0, the second member would tend to the limit j4, 

 which by hypothesis is different from ; and yet, from the na- 

 ture of exponential functions, the limit of the first member is zero. 



We conclude, therefore, that the function e cannot be deve- 



loped in a series of the kind assumed, although it vanishes with its 

 variable ; and consequently that, if we only know this property of a 

 function, that it vanishes when its variable vanishes, we cannot cor- 

 rectly assume that it may be developed in such a series. 



If any doubt should be felt respecting the truth of the remark, 



that the function x e tends to zero along with ^r, when a is 



any positive constant, this doubt will be removed by observing that 



the function x e , which is the reciprocal of the former, increases 

 without limit while x decreases to zero. For we may develope this 

 latter function, by the known theorems, in the essentially converg- 

 ing series, 



xe =y e zzy +y + •■'_^_ + JL^__ -|- L_ + &c., 



y being the reciprocal of x ; and while y tends to -j- as , the terms of 

 this series remain all positive, and all after a certain constant num- 

 ber increase indefinitely. 



