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
$$\lim_{x\rightarrow a}f(x)^{g(x)}\not=\lim_{x\rightarrow a}\ln\left(f(x)^{g(x)}\right).$$

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 $$\lim_{x\to 1}\frac{\sin\left(\frac{\pi}{2}x\right)\ln(2-x)}{\cos\left(\frac{\pi}{2}x\right)}.$$