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Limit

equation, roots, positive and values

LIMIT, in a restrained sense, is used by mathematicians for a determinate quan tity to which a variable one continually ap proaches ; in which sense the circle may be said to be the limit of its circumscrib ed and inscribed polygons. In algebra, the term limit is applied to two quanti ties, one of which is greater, and the other less, than another quantity ; and in this sense it is used in speaking of the limits of equations, whereby their solution is much facilitated.

Let any equation, as x3 -p x3 x q x r = o be proposed ; and transform it into the following equation : y3 +3 e y' + 3 e. y + el Śp 2p e y Śp 8. 0 .

r Where the values of y are less than the respective values of x, by the difference e. If you suppose e to be taken such as to make all the coefficients of the equation y positive, viz. e3 q eŚ r, 3 e' Ś2p e + q, 3 then, there being no variation of the signs in the equation, all the values of y must be negative ; and consequently the quantity e, by' which the values of x are diminished, must be great.

er than the greatest positive value of x; and, consequently, must be the limit of the roots of the equation x3 Śp x3 + q x r = o.

It is sufficient, therefore, in order to find the limit, to inquire what quantity substituted for x, in each of these expres sions s3 Ś p xx+ g x Ś r, 3 Ś 2 px + q, 3 xŚp, will give them all positive ; for the quantity will be the limit required.

Having found the limit that surpasses the greatest positive root, call it m. And if you assume y = m Ś x, and for x sub stitute m Śy, the equation that will arise will have all its roots positive ; because m is supposed to surpass all the values of x, and consequently x (= y) must always be affirmative. And,by this means, any equation may be changed into one that shall have all its roots affirmative.

Or, if Ś n represent the limit of the ne gative roots, then by assuming y = x n the proposed equation shall be trans formed into one that shall have all its roots affirmative ; for n being greater than any negative value of x, it follows that y = x n must be always positive.

What is here said of the above cubic equation, may be easily applied to others ; and of all such equations, two limits are easily discovered, viz. 0, which is less than the least ; and e, found as above, which surpasses the greatest root of the equa tion. But besides these, other limits still nearer the roots may be found ; for the method of doing which, the reader may consult Maclaurin's Algebra.