Vtyjkyuk

I was looking at the area under $y=1over x$, and randomly I wondered if there was some point at which the areas on either side were equal.

I set about finding the answer using this method:
$$\
lim_ato0^+ int_a^pdxover x=lim_btoinfty int_p^bdxover x\
lim_ato0^+ln p - ln a=lim_btoinftyln b - ln p\
2ln p=lim_ato0^+ln a+lim_btoinfty ln b\
2ln p = -infty+infty\
$$
Upon seeing the last line, I concluded that given the indeterminate form, $pin(0,infty)$, which I interpreted to mean "$p$ does not exist"

Is this rigorous enough, or did I skip some steps?

We have that $lim_ato0^+ int_a^pdxover x$ diverges and $lim_btoinfty int_p^bdxover x$ diverges too. So, since these two areas tend to $infty$, you cannot compare those two. Hence you cannot find a point where these two areas are equal.

From the indeterminate that you found you cannot conclude the non-existence of $p$ cause an indeterminate form of $infty -infty$ with some manipulations can lead to a number or not.

Your statement that $p$ does not exist is correct.

You know that the total integral is $infty$ and you are looking for a point which divides infinity in two equal parts.

Now if the point $p$ is a real point,then the first half of the integral will be bounded and can not be infinity.

Thus even without integration we notice that such a point does not exist.

The same argument goes for every integral which diverges to infinity.