. A new treatise on the elements of the differential and integral calculus . ^ constant , Xow, as z diminishes without limit, az Avill also diminishwithout limit; and therefore lim. (1 -[- az) =^ e ; lim. (l -\^ az)^ = e«.12. In any system of logarithms, z lim. —^—i—! — lim. Z( I + zY = Le; z - I - and, if the logarithm be taken in the Napierian System, l™. Ki+l) = /« = !. z 13, Resuming the equation 2 and making 1 -\- z =^ a^, whence (taking logaritlims in tlie sys-tem of wliich a is the base), r r= Z(l -h c) and c : - a— 1 ;therefore L{\ +■-)= V a— 1 or, by taking the reciprocal
. A new treatise on the elements of the differential and integral calculus . ^ constant , Xow, as z diminishes without limit, az Avill also diminishwithout limit; and therefore lim. (1 -[- az) =^ e ; lim. (l -\^ az)^ = e«.12. In any system of logarithms, z lim. —^—i—! — lim. Z( I + zY = Le; z - I - and, if the logarithm be taken in the Napierian System, l™. Ki+l) = /« = !. z 13, Resuming the equation 2 and making 1 -\- z =^ a^, whence (taking logaritlims in tlie sys-tem of wliich a is the base), r r= Z(l -h c) and c : - a— 1 ;therefore L{\ +■-)= V a— 1 or, by taking the reciprocals, 1 a —1 1 X(l +Zr: V Now, as c dimiiushes without limit, so also will /•, and they willreach the limit zero tog(4hor : (herc(i>ro 20 DIFFERENTIAL but lim. V Suppose a = e, whence m=^la;and therefore g mv -j^ lim. = Ze=:m. V 14, To define some of the terms, and explain the meaningof some of the symbols, employed in the calculus, let us takethe explicit function of a single variable and give to x an increment denoted by ax ; y will receive thecorresponding increment Ay= a/(^) =:/(^ + ax)—/(a?), and therefore A^ ^ f{^j:J^^x)—f{x)^ AX AX When AX = 0, the ratio ■— takes the form tt : yet it has in AX ^ * fact a determinate value, which is generally some other func-tion of cc, and expresses, as will be seen presently, the tangentof the angle that a straight line, tangent to the curve of whichy =f[x) is the equation, makes with the axis of the variableX. This limiting value of the ratio of the increment of thevariable to the corresponding increment of the function iscalled the differential co-efficient, or derivative of the func-tion, and is represented by the notations / r// X ^^ r ^^^ V f{x^Ax) — f(x)y fix), -r- , lim. —newtreatiseonele00robi
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