Methods for computing limits of functions given by formulas that cease to
have a meaning when the limiting values of the argument are
formally substituted in them, that is, go over into expressions like
for which one cannot judge whether the required limits exist or not
without saying anything about finding their values if they exist.
The basic instrument of evaluating indeterminacies is Taylor's formula (cf.
Taylor formula),
by means of which one singles out the principal part of a
function. Thus, in the case of an indeterminacy of the type
,
for which in order to find the limit
where
one represents the functions
and
by Taylor's formulas in a neighbourhood of
(if this is possible) up to the first non-zero term:
as a result one finds that
In the case of an indeterminacy of the type
,
in order to find the limit
where
one applies the transformation
which reduces the problem to the evaluation of an indeterminacy of type
.
Indeterminacies of the types
or
are also conveniently reduced to type
by the following transformations:
respectively.
For evaluating indeterminacies of the types
,
or
it is appropriate first to take the logarithm of
the expressions whose limits are to be found.
Another general method for evaluating indeterminacies of the types
or
and those reducible to them is the
l'Hospital rule.