Indeterminate Powers

An indeterminate power is what we call the form of a limit $\displaystyle\lim_{x\rightarrow a}f(x)^{g(x)}$ where the limit has the form $0^0$ or $1^\infty$ or $\infty^0$ when you plug in the value $a$ (which might be $\infty$).

Recall that when we have $f(x)^{g(x)}$ and we wish to differentiate it, we use logarithmic differentiation, since taking the log of $f(x)^{g(x)}$ allows us to pull the power $g(x)$ down so it is easier to deal with.

Similarly, when looking at  $\displaystyle\lim_{x\rightarrow a}f(x)^{g(x)}$, we can take the log of $f(x)^{g(x)}$, and turn this into an indeterminate product, which we can then tackle with L'Hospital's rule.  We have to be careful, as you will see in the video, since

The video below will explain in detail how we compute such limits.

Warning:  There is an error in the last step of the last example on the video, when she uses L'Hospital's rule on