which, when x = a, is either O, e-a, or ∞, according as n is greater than, equal to, or less than, 1.

On examination of the above examples, it will be seen that the result is infinitesimal, finite or infinite, according as the order of infinitesimals in the numerator, determined by the method of Section 1 of this Chapter, is higher than, the same as, or lower than, the order of the denominator.

125.] Evaluation of indeterminate quantities of the form. Let f(x) and (x) be two functions of x, which become infinite when x = xo; then, as their reciprocals become infinitesimals, the ratio of the functions may be evaluated by the former method, as follows;

whence, dividing out common terms, we have, when x = x。,

If the first-derived-functions f'(x), '(xo) also become infi

nite, their ratio must be evaluated in the same way as

and we shall have

f(x)

(xo)

and if the several derived-functions vanish or, if it is possible, become infinite up to the nth, when x = x。, and the nth are finite,

f(xo) f'(xo)

=

$(xo) φ' (20)

fn (xo)
φ" (πο)

The determinate value therefore of such indeterminate functions is the ratio of those derived-functions of the numerator and denominator which first become finite when x = xo.

The result of which shews that e* is, when x = ∞, an infinity of a higher order than "; also a similar property is true of a*; a finite quantity therefore raised to an infinite power is an infinity of a higher order than the same infinite quantity raised to a finite power.

1

∞

; so that z = ∞, when x = 0; and the func

therefore the function is equal to 0, when x = 0; whence we

1

conclude that e a2 is an infinitesimal of a higher order than x2", however large be the value of n.

This is a result of some importance; insomuch as it shews

1

that although evanishes, when x = 0, whence it might be inferred that x is a factor of it, yet it admits of no factor of the form 2" however large n be. Compare equation (23), Art. 116, and the remarks thereupon.

if x therefore becomes infinite, it is an infinity of a higher order than its logarithm.

126.] Evaluation of functions which for particular values assume the indeterminate form 0 × ∞.

Let f(x) and (x) be two functions of x which respectively become O and ∞, when x = xo; then their product may be put under the form

'(x), which = 0, when a = x。,

1

$(x)

and may be evaluated according to the method of the last two Articles.

Ex. 1. Evaluate e- log r, when x = x.

127.] Evaluation of functions which for particular values assume the form o

3.

Let f(x) and p(x) be two functions of x, which = 0, when

Ex. 2. Evaluate sec - tan x, when x =

π

2

π

2

π

therefore, when x =

2

sec x and tan x are either absolutely

equal, or differ by a quantity which must be neglected in their

SECTION 3.-Evaluation of Explicit Functions of a single variable, which for particular values assume the forms 0o, ∞o, 1°, 0°.

128.] Let f(x) and (c) be two functions of a which, when x = x。, assume such values that f(x) is of one or other of the above forms.

and as log f(x) has singular values when f(x) = 0, or = 1, or = ∞, we may express the last equation in the form

which, for the particular value of a。, will be of the form or

∞

, and may be evaluated according to the methods explained in the last Section.