Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 9]_ V Hospital-Bernoulli Rule for Indeterminate Forms_79
The rule is also applicable when a = 00.
If the quotient
/ fix)
,, again yields an indeterminate form, at the point
x = a, of one of the two above-mentioned types and /' (x) and q>' (x) satisfy
all the requirements that have been stated for f(x) and q? (x), we can then
pass to the ratio of second derivatives, etc.
However, it should be borne in mind that the limit of the ratio -^-~
may exist, whereas the ratios of the derivatives do not tend to any limit
(see Example 809).
2. Other indeterminate forms. To evaluate an indeterminate form like
0oo, transform the appropriate product fi(x)*ft (x), wnere lim/, (jt) = and
K+O.
/(*)
(T^T\ (the form -).
= lim/2 (*) oo, into thequetient ^^ (the form -
*->a *
M*)
U /i (X) oo
In the case of the indeterminate form oo oo, one should transform the
appropriate difference /,(*) f 2 (x) into the product / t (x) l and
L / 1 (x )j
first evaluate the indeterminate form 7*7^;
duce the expression to the form
/Tw
if lim 7^7^=1, then we re-
r i (X) x-+a i\ \x )
(the form ).
The indeterminate forms I, 0, 00 are evaluated by first faking loga
rithms and then finding the limit of the logarithm of the power [fl (x)]^ (x}
(which requires evaluating a form like 0oo).
In certain cases it is useful to combine the L'Hospital rule with tht
finding of limits by elementary techniques.
Example 1. Compute
Solution. Applying the L'Hospital
lim JL1 (form ").
*->o cot x oo 7
rule we have
lim JEfL^llm pL*r lim .
x+ocotx jc-o(cot*) jc-*o x
We get the indeterminate form -jp however, we do not need to use the
L'Hospital rule, since
We thus finally get
Um
sint *
Hm
sin *
C-frO X ~~*-H) X
"
JC->0 COt X