Зображення сторінки
PDF
[merged small][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors]

(34) U = 2" log x when x = 0, n being positive.

[ocr errors][ocr errors]

and therefore by the last example

u = 2" log x = 0, when x = 0.

[merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors]

Differentiating three times, we find the real value to be

[ocr errors]

which is the sum of the reciprocals of the squares of the natural numbers.

(38) The sum of the series
II
1? + x 327

. + &c. to

[ocr errors]
[merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors]

which is therefore the sum of the squares of the reciprocals of the odd numbers.

The reader will find other examples of a similar kind relating to the summation of series in Euler's Calc. Diff: p. 760, seq.

Sometimes the value of an indeterminate function may be most readily found by throwing it into a form in which its real nature is more easily seen.

(39) If u = 24 sin find its value when x = 0 .

[merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small]

(10)

[ocr errors]

12 lata

} = 0.00, when

[merged small][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][subsumed][ocr errors]

Functions which for a particular value of the variable take the form 0° 0° 1c, may be reduced to a shape in which the preceding methods are applicable. Let x and y be functions of w and u = *", then if for x = a

[ocr errors][ocr errors]

Now since % = elog?, u = eyloga ; and these three cases are reduced to the determination of y log %, which takes the form 0 x $o.

De Morgan's Diff: Calc. p. 175. (42) Find the value of 2020 when x = 0, a being positive. This is equivalent to fita loga, and we have to find the value of 219 log x when x = 0. Now by Ex. (34) 29 log x = () when x = 0. Therefore

zita = e = 1 when x = 0. If a be negative 2020 = 0) when x = 0). (13) Find the value of

u= C) = 6" when x = 0.

ma

[ocr errors][ocr errors]

(44) u = pisin x when x = 0.
We may arrive at the value of this function by the

sin x consideration that, when x is indefinitely diminished, — =1, or sin x = x : therefore when x = 0, æsin x = level = 1, by Ex. (42). In the same way it would appear that

(sin x) sin 2 = 1 when x = 0.

[ocr errors][ocr errors]

it appears that sin x . log x = 0 when x = 0); and similarly that sin x. log (sin x) = () when x = 0.

(45) u = (cot x)SİN 2 = 60" when x = 0.
By similar considerations it appears that

(cot x) sin ! = 1 when l= 0).

[merged small][ocr errors]

log (1 +- nr) Here u = e , and we have to find the value of log (1 + nx)

? when w = 0. Differentiating we find this to

[ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors]
[ocr errors]

This result may be verified by expanding u by the binomial theorem: that gives

1 1 na 202 1 / 1 u=l+

1.2 1

11.2.3

[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors]
[ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][ocr errors][merged small][ocr errors][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small]

Functions which for a particular value of the variable take the form o', have been used by Libri to introduce discontinuity into ordinary functions.

Thus, if it be desired to express a function f (x) which shall be equal to (w) from x = -00 to « = n, and to y (x) from X = n to x = 0, he writes

f(x) = (1 - 004") 0 (0) +00** ¥ (w). See his Mémoires de Mathématique et de Physique, Vol. 1. p. 44, and Crelle's Journal, Vol. x. p. 303. In the same Journal, Vol. x11. p. 134 and p. 292, the reader will find some discussion on the real value of this indeterminate expression,

« НазадПродовжити »