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374 Rh x-a appearing in both numerator and denominator. Now we cannot divide by a zero factor, but as long as x is not absolutely equal to a, the factor x-a may be removed, and we then find that the nearer x approaches to the value a, the nearer does the value of the fraction approximate to 3 2, or in accordance with the definition of Art. 458,

-, .1 T .. n x^ -^ax — 2 a^. 3 when x = a, the limit oi ^ is -• x^ — a- 2

464. Vanishing Fractions. If f(x) and f'(x) are two functions of x, each of which becomes equal to zero for some particular value a of x, the fraction f(a) f'(a) takes the form 0 0, and is called a vanishing fraction.

Ex. 1. If x = 3, find the limit of x-^ — X"^ — ox

When x = 3, the expression reduces to the indeterminate form 0 0; but by removing the factor x-3 from numerator and denominator,

^x + 1 ^j^g^^ ^ x2 4- 2 X + 1

which is therefore the required limit.

Ex. 2. The fraction y^ — Vx + a ^y^^Q^^Q^ ^^^^^^ ^^^ X — a

To find its limit, multiply numerator and denominator by the surd conjugate to 3x-a - x + a ; the fraction then becomes

(?, X - a) - (X -{- a) 2 (x — a) ( V3 X — a + Vx + a) /3 x — a + Vx + a

whence by putting x = a we find that the limit is 1 2a.

Ex. 3. The fraction becomes 0 0 when x = 1.

To find its limit, put x = 1+h, and expand by the Binomial Theorem. Thus the fraction

1-(1 + /.)^ l-Cl+l/^-^A^ + .-V -.-r..

Now h = 0 when x = 1; hence the required limit is 5 3.