the other terms; that is, we may take the first term n^4 as the equivalent of the whole expression, with an error as small as we please provided 2 be taken large enough.
Ex. 2. Find the limit of when (1) x is infinite; (2) x is zero.
(1) In the numerator and denominator we may disregard all terms but the first ; hence limit 382% —20?—4 _ 32% _3
=O 5x3 4e48 5ad 5
(2) When x is indefinitely small we may disregard all terms but the last ; hence the limit is a -4 8 or -1 2.
VANISHING FRACTIONS.
463. Suppose it is required to find the limit of vV+ax—2¢ ew when x = a.
If we put x=a+h, then h will approach the value zero as x approaches the value a. Substituting a+h for x,
etaa—2a_ 38aht+l? _3a+h. eo 2uaht+h 2at+h’
and when h is indefinitely small the limit of this expression is 3 2.
There is, however, another way of regarding the question; for
e+ ae—20 _w-—a@+2a)_2+26 a na a)(c omen)
and if we now put x=a the value of the expression is 3 2, as before.
If in the given expression we put x=a before simplification, it will be found that it assumes the form 0 0, the value of which is indeterminate [Art. 183]; also we see that it has this form in consequence of the factor
