What do we do when we try to evaluate a limit and we end up with one of
indeterminate powers
1^{∞},
0^{0}, or
∞^{0}?

The standard procedure is to change this power into a product using the
"*e* to logarithm" trick. This trick is based on the fact that for every
positive real number *x* we have
*x* = exp(ln(*x*))

*A*^{B} =
*e*^{ln(AB )} =
*e*^{B⋅ln(A )}.

Since this is just an algebraic transformation, the limit of the expression
on the left is the same as the limit of the expression on the right. That one
has the form of "*e* to something". Since exponential is a "nice
function", we can "pull it out of the limit" and find the limit of the
exponent first. This expression is a product, but what kind?
If we start from an
indeterminate power, we always get an indeterminate product:

We actually cheated a bit in the last case, see the note below.

Anyway, we transformed the problem of finding a limit of an indeterminate
power into a problem of finding a limit of an indeterminate product.
There we can apply the standard procedure (change the product into a ratio,
then most likely use l'Hospital, see the box
"indeterminate product"). After we
find the limit of the exponent, we have to **remember to put it back into
the exponential.**

**Example:**

Now we need to find the limit of the product
*x*⋅ln(*x*)

Note that this transformation can be actually used anytime. For instance, if
you do not remember (or trust) that in the limit algebra
^{∞} = 0,

^{∞} =
*e*^{∞ln(1/2)}
= *e*^{−∞}
= 0.

(We used the fact that

**Note:** An inquisitive reader should have gotten suspicious when we were
transforming the power 0^{0} using the "*e* to ln" trick. Why?
Because ln(0) does not exist, we cannot plug zero into the logarithm. What is
happening here? The secret is that we cheated a bit when writing the power.
General power requires that the base be positive, so in fact we do not have
0^{0}, but ^{+})^{0}.^{+})

Why didn't we write the proper expression
^{+})^{0}*always* check on 0 when putting it into logarithm in the limit algebra,
and if the zero in the power were not of the positive type,
we would find out anyway when working with logarithm and realize that there is
something fishy.

In Solved Problems - Limits, these methods are used in this problem and this problem.

Next box: comparison and oscillation

Back to Methods Survey - Limits