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INDETERMINATE FORMS When the value of a limit is obtained by substitution and
it assumes any of the following forms, another method for finding the limit
must be used:
These are called indeterminate forms. There are many methods of evaluating indeterminate forms.
Two methods of evaluating indeterminate forms are (1) factoring and (2)
division of the numerator and denominator by powers of the variable. Sometimes
factoring will resolve an indeterminate form. EXAMPLE: Find the limit of
SOLUTION: By substitution we find
which
is an indeterminate form and is therefore excluded as a possible limit. We must
now search for a method to find the limit. Factoring is attempted, which
results in
so
that
and
we have a determinate limit of 6. Another
indeterminate form is often met when we try to find the limit of a function as
the independent variable approaches infinity. EXAMPLE: Find the limit of
SOLUTION:
If we let x approach infinity in the original expression, the result will be
which must be excluded as an indeterminate form. However,
if we divide both numerator and denominator by x^{4}, we obtain
and we have a determinate limit of
. PRACTICE PROBLEMS: Find the limit of the following:
ANSWERS:
