What does the $L^p$ norm tend to as $p\to 0$? [duplicate]

Assume the limit exists and call it $L$. Then $$\log L = \lim_{p\to 0}\frac{\log \left( \int_a^b f(x)^p dx \right)-\log (b-a)}{p},$$

which is just $$\left.\frac{d}{dp}\right|_{p=0}\int_a^b f(x)^p dx=\int_a^b \log f(x) dx, \quad \text{so}\\L = \exp\left[ \int_a^b \log f(x) dx\right].$$

This jibes with what we know from the power means, because $\exp\left(w_1 \log a_1 + \dotsb + w_n \log a_n\right)$ is just the geometric mean.

More questions: are the $p$ norms in ascending order of $p$ like the power means are?

Since you normalized the measure space to have total measure $1$, Jensen's inequality immediately implies the monotonicity of $L^p$ norm (quasinorm for $p<1$) with respect to $p$. Namely, for all $p>0$ and all $r>1$ we have (using the convexity of $t\to t^r$ on $[0,\infty)$)
$$ \left(\int_X |f|^{p}\right)^{r} \le \int_X (|f|^p)^r $$ hence $\|f\|_p\le \|f\|_{pr}$.

You can also use Hölder's inequality to the same purpose.

Thus, the limit $\lim_{p\to 0}\|f\|_p$ exists under the rather weak assumption that some $L^p$ norm of $f$ is finite. As you noted, it is equal to $\exp\left(\int_X \log|f| \right)$. Continuity and the particular form of measure space are irrelevant.