L'Hopital's Rule: x to the power x: Find

.

If we were to substitute directly, we would have the form . Hence, we can't yet apply L'Hopital's Rule. Let's try taking the logarithm of both sides:

. Since the natural logarithm function is continuous, we can rewrite the equation as

. Next we use the properties of logarithms to get

. Here the form is . We need one more change to get a ratio:

we now have the form and can apply L'Hopital's Rule. That is, we differentiate the numerator and the denominator separately, take the ratio, and evaluate the limit of the resulting expression. (We do not use the Quotient Rule).

Applying this rule, we get

Carrying out the differentiation, we get

. If the ln L is 0, then L itself must be 1. Consequently, we conclude that L, our original goal, is 1.