Ask Uncle Colin: A Limiting Issue

Dear Uncle Colin,

I have a problem with a limit! I need to figure out what ${(\tan \left(x\right))}^x$ is as $x\to 0$.

-- Brilliant Explanation Required Now! Our Understanding’s Limited; L’Hôpital’s Inept

Right, BERNOULLI, stop badmouthing L’Hôpital and let’s figure out this limit. It’s clearly an indeterminate form to begin with, as as $x$ gets small, $(\tan (x){)}^x$ approaches $0^0$, which isn’t defined.

So let’s do something clever. If we call the limit $L$ and take logs, we get:

$\ln (L)=x\ln (\tan (x))$, which is again indeterminate – although now it’s (loosely) $0\times \mathrm{\infty }$, which isn’t defined.

I’m going to use one of my favourite tricks, which is to rewrite $\tan (x)$ as $x\times \frac{\tan (x)}{x}$, making the log of the limit:

$\ln (L)=x\ln (x)+x\ln \left(\frac{\tan (x)}{x}\right)$

Applying L’Hôpital’s rule to $\frac{\tan (x)}{x}$ gives a limit as $x\to 0$ of $\frac{\sec (x)}{1}$, which goes to 1, meaning that the second term goes to $0\times 0$ and can be ignored.

The first term can be rewritten as $\frac{\ln (x)}{1/x}$, another indeterminate form we can apply L’Hôpital to – the limit becomes $\frac{\frac{1}{x}}{\frac{-1}{x^2}}=-x$, which again goes to 0 as $x\to 0$.

So, in the limit as $x\to 0$, $\ln (L)\to 0$, and as a result, $L\to 1$.

Hope that helps!

-- Uncle Colin