L'Hopital's Rule
Other "Indeterminate Forms"
" " is called an indeterminate form because knowing that
approaches
and
approaches
is not enough to determine the limit of
, even if it has a limit. The ratio of a "small" number divided by a "small" number can be almost anything as the three simple "
" examples show:
Similarly, " " is an indeterminate form because knowing that
and
both grow arbitrarily large is not enough to determine the value limit of
or if the limit exists:
Besides the indeterminate quotient forms " " and "
" there are several other "indeterminate forms". In each case, the resulting limit depends not only on each function's limit but also on how quickly each function approaches its limit.
Product: If approaches
, and
grows arbitrarily large, the product
has the indeterminant form "
".
Exponent: If and
both approach
, the function
has the indeterminant form "
".
If approaches
, and g grows arbitrarily large, the function
has the indeterminant form "
".
If grows arbitrarily large, and
approaches
, the function
has the indeterminant form "
".
Difference: If and
both grow arbitrarily large, the function
has the indeterminant form "
".
Unfortunately, l'Hô pital's Rule can only be used directly with an indeterminate quotient (" " or "
'), but these other forms can be algebraically manipulated into quotients, and then l'Hô pital's Rule can be applied to the resulting quotient.
Example 5: Evaluate ("
" form)
Solution: This limit involves an indeterminate product, and we need a quotient in order to apply l'Hô pital's Rule. We can rewrite the product as the quotient
, and then so apply l'Hô pital's Rule
A product with the indeterminant form "
" can be rewritten as a quotient,
or
, and then l'Hô pital's Rule can be used.
Solution: An indeterminate exponent can be converted to a product by recalling a property of exponential and
logarithm functions: for any positive number so
.
and this last limit involves an indeterminate product
which we converted to a quotient and evaluated to be
in Example 5.
Our final answer is then :
An indeterminate form involving exponents, with the form "
, " "
, " or "
, " can be converted to an indeterminate product by recognizing that
and then determining the limit of
. The final result is
.
Example 7: Evaluate ("
" form)
Solution: so we need
"
" an indeterminate product so rewrite it as a quotient
an indeterminate quotient so use l'Hô pital's Rule