B. The calculation of certain limits. Suppose that and are two functions of whose derivatives and are continuous for and that and are both equal to zero. Then the function is not defined when . But of course it may well tend to a limit as .
Now where lies between and ; and similarly , where also lies between and . Thus We must now distinguish four cases.
(1) If neither nor is zero, then
(2) If , , then
(3) If , , then becomes numerically very large as : but whether tends to or , or is sometimes large and positive and sometimes large and negative, we cannot say, without further information as to the way in which as .
(4) If , , then we can as yet say nothing about the behaviour of as .
But in either of the last two cases it may happen that and have continuous second derivatives. And then where again and lie between and ; so that We can now distinguish a variety of cases similar to those considered above. In particular, if neither second derivative vanishes for , we have
It is obvious that this argument can be repeated indefinitely, and we obtain the following theorem:
suppose that and and their derivatives, so far as may be wanted, are continuous for . Suppose further that and are the first derivatives of and which do not vanish when . Then
(1) if , ;
(2) if , ;
(3) if , and is even, either or , the sign being the same as that of ;
(4) if and is odd, either or , as , the sign being the same as that of , while if the sign must be reversed.
This theorem is in fact an immediate corollary from the equations
Example LVIII
1. Find the limit of as . [Here the functions and their first derivatives vanish for , and , .]
2. Find the limits as of
3. Find the limit of as . [Put .]
4. Prove that being any integer; and evaluate the corresponding limits involving .
5. Find the limits as of
6. , , as .