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The value of this fraction, when x = 0, is the same as that of

(€* – 6--)

ima

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The value of u will in this case coincide with that of

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P

S

Q = (2a)} - (v + a)] + (x – a)?:

(x – b)}+ }(x a) (x b)-S + } (x – a)-), Q = -} (x + a)-} + } (x – a)-1.

P (x b)! + } (x – a) (x – b)-€ + (x – a)-}
Hence u =
Q

- 4(x + a)- + }(x - a)-!

or, multiplying numerator and denominator by 2 (x – a),

2 (x – a)(x 5)2 + (x – a)/(x b)-} + 1

- (x - a) (x + a)-5 + 1

=l.

The following process may frequently be applied with advantage to the evaluation of indeterminate fractions : “ If

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x = a cause the fraction to assume the form

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a + h for v in both numerator and denominator, and develop both according to powers of h; reduce the new fraction to its simplest form, and then make h = 0; the result will be the true value of the fraction.” It is easily seen that this includes the ordinary method of differentiation.

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Since in this case u would be impossible if x were greater than a, assume x = a h; the result is

(2a)?

when h 0, 1+ 34a

3a'

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Let x = 0) + h, or h; then, expanding the circular function by the formula

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27?h? T+

1.2.3

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when h = 0.

2hh +

2 (77h)
1.2. 3.

+ &c

+

1

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There are other forms of functions which become indeterminate for a particular value of the variable, but these may generally be reduced to the form Thus, if u = P. Q, and

0

which is

if for x = a, P = 0, Q = 0, we have u = ( x 00,

1 indeterminate. But if we assume Q

Q
Р 0

when x = a.

we have

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U =

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U =

Qi'

Therefore, applying the rule for vanishing fractions,

P

P
Q

when x = a.

Q'' But if Q = c when x = a, all its differential coefficients will also be infinite, and u, taking the form , is still indeterminate unless the factor which becomes infinite should happen to divide out.

Р
To determine the value of a function u = which

Q

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Treating this as a vanishing fraction,

P Q

when X = a.
P'

P P'
Whence U =

when q = a.

QQ''
But if P
oo, and Q

= a, all their differential

P' coefficients will also become infinite, and will still have

Q'

= 0, for a

the form. This reduction, then, will not give us the true value of the fraction unless some factor divide out, or we can find some relation, depending on the nature of the functions, between the new numerator and denominator which will enable us to trace the real value.

A function u = :P Q, which becomes co co when x = a, can frequently be reduced to the form for if P and

P

1

1

1

Q

=

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1

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when x = ll,

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Q

Qi - P

PQI 0 and its value is to be found by the usual method.

De Morgan's Differential Calculus, p. 172.

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Differentiating the numerator and denominatorn times in succession, n (n-1)...2.1

when x = 00 ,

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