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Evaluation of Functions of the form %.

64. The rule for the evaluation of functions of x, which for particular values of the variable assume the form %, is applicable also for the evaluation of functions which assume the form.

Let

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Hence, by the theorem of Art. (63), for this value of x,

=

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If f'(x) = ∞ and F(x)= ∞, then, by the application of

the same principle, it is plain that, for this particular value of x,

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where + 0 denotes the limit of positive magnitude.

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It would be more easy, in order to evaluate (x), first to find the value of

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a, m, n, being positive quantities, and a being greater than

unity.

If we differentiate the expression

Хт a*

m times above and below, the series of resulting fractions will

always be of the form , up to the mth differentiation, which will give a fraction

C

+(x)

C being finite and ¥ (x) infinite, when x = ∞: thus we see that

(∞) = 0.

Failure of the method of Differentials for the Evaluation of Indeterminate Functions.

65. It occasionally happens, when, for a particular value of x,

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that the application of the above rules is inadequate for the evaluation of (x). By virtue of these rules we may take

ƒf'(x)

as an equivalent for f(x).

; we may modify

f'(x) F'(x)

as

F(x)

F'(x) we please by cancelling or introducing common factors above

and below; and, supposing (2)

to represent the modified

F(x)

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and so on indefinitely, until we at length arrive at a fraction of which at least both the numerator and denominator are not simultaneously zero or simultaneously infinite. Sometimes, however, we are unable by the application of these combined operations to rescue the function from its nugatory form, the indeterminateness perpetually presenting itself again and again in the successive functions either in the shape of or of Ro. Ex. Suppose that

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0

and that x = ∞ and that it is required to find the value of

(x) for this value of x, m and n being positive numbers, and both a and b being greater than unity.

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We shall fall into the same difficulty at each succeeding operation. In fact, the method of differentiation must not be considered as a universal rule for evaluating indeterminate functions, but merely as an instrument frequently of great use for this purpose.

Evaluation of Indeterminate Functions of several Independent Variables.

66. Suppose that, to take the case of two independent variables,

$(x, y) =

f(x, y) _ 0
F(x, y) ō

=

when x, y, receive respectively particular values x,y; the variables x and y being subject to no equation of connection. Generally

p(x + dx, y + dy) = F(x, y) + ▲ F(x, y)'
f(x, y) + ▲F(x, y)

▲ƒ(x, y) and ▲ F(x, y) denoting the total differences of f(x, y) and F(x, y).

Suppose that x = x, y = y: then

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supposing that, in the expressions for Af(x, y), ▲F(x, y), which are certain functions of x, y, dx, dy, we finally substitute

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