. Differential and integral calculus, an introductory course for colleges and engineering schools. P remain fixed, while Q traverses the curve in such a way as to approach P as its limiting posi-tion. Then the secant PQ will turnabout P as a pivot, and will approacha limiting position, TP, and thislimiting position we define to be thetangent to the curve at P. The tangent is sometimes de-scribed (though inaccurately) as aline that joins two infinitely near points of the curve. It is plainthat, as Q approaches P, the angle a which the secant makes withOX approaches, as a limit, 6, th
. Differential and integral calculus, an introductory course for colleges and engineering schools. P remain fixed, while Q traverses the curve in such a way as to approach P as its limiting posi-tion. Then the secant PQ will turnabout P as a pivot, and will approacha limiting position, TP, and thislimiting position we define to be thetangent to the curve at P. The tangent is sometimes de-scribed (though inaccurately) as aline that joins two infinitely near points of the curve. It is plainthat, as Q approaches P, the angle a which the secant makes withOX approaches, as a limit, 6, the angle which the tangent makeswith OX. 20. General Theorems of Limits. We shall now state somegeneral theorems of limits, some of which we have made tacituse of in the foregoing pages. These theorems admit of rigorousproof, but these proofs do not belong to an elementary course inthe calculus. In fact, most of these theorems are fairly obviousin the case of the simple functions that we shall deal with. a. The limit of f(x) is independent of the law of variation bywhich x approaches its §20 LIMITS 29 This means that whether x approaches its limit in such a waythat the difference x — a (or a — x) has the series of values(Art. 9, example 1), 2d> 2>d> 2>d> * *or the series of values (Art. 9, example 3), 1j 1a 1a -d, -d, jd, . . or some other series of values, f(x) has in each case the same limit,or if f(x) has no limit in one case it has none in any other are exceptions to this principle, some of which are illustratedin the following examples. (1) Lim = +oo or — oo , according as x in approaching a is x—a X (Z always greater or less than (2) Lim Cx~a = oo or 0, according as x remains greater or less x=a than a. (3) Lim tan x = + oo or — oo, according as x remains less or 7T greater than - (see Art. 17).A 13. If two functions of the same argument are equal for all valueswhich that argument takes in approaching a prescribed limit, the
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