Because of G36's comments/questions below this answer, of late, I'd editing this answer to include the development of the equation I provided at the outset. It's not complicated.
We start out with the simple KVL equation:
$$V_\text{CC}-I_\text{LED}\cdot R_\text{LIMIT}-V_\text{LED}=0\:\text{V}$$
And solve it for \$I_\text{LED}\$:
$$I_\text{LED}=\frac{V_\text{CC}-V_\text{LED}}{R_\text{LIMIT}}$$
Now, our goal is to compute sensitivity equations from the above. A sensitivity equation just quantifies the output uncertainties based upon the effects of input uncertainties. I didn't find any really easy paper on the topic, but I did find a reasonably readable one here: Sensitivity Analysis for Uncertainty. So, feel free to read that if you have any doubts about the rest of what I write, below.
We want to find the % variation of something with respect to the % variation of something else. In calculus form, % variation looks like \$\%\,x = \frac{\text{d}\,x}{x}\$. This is the exact % variation, which is much better than the finite approximation variation that is \$\%\,x \approx \frac{\Delta\,x}{x}\$. It turns out that the calculus mindset is actually not so hard to do.
First, we apply the implicit product rule (or multivariate chain rule):
$$\text{d}\,I_\text{LED}=\frac{\text{d}\, V_\text{CC}}{R_\text{LIMIT}}-\frac{\text{d}\, V_\text{LED}}{R_\text{LIMIT}}$$
The we divide both sides by \$I_\text{LED}\$:
$$\begin{align*}\%\, I_\text{LED}=\frac{\text{d}\,I_\text{LED}}{I_\text{LED}}&=\frac{\text{d}\, V_\text{CC}}{R_\text{LIMIT}\,I_\text{LED}}-\frac{\text{d}\, V_\text{LED}}{R_\text{LIMIT}\,I_\text{LED}}\\\\&=\frac{\text{d}\, V_\text{CC}}{V_\text{CC}-V_\text{LED}}-\frac{\text{d}\, V_\text{LED}}{V_\text{CC}-V_\text{LED}}\end{align*}$$
Now, we need to convert the infinitesimals on the right side into % variations. This is simple to do:
$$\begin{align*}\%\, I_\text{LED}&=\frac{\frac1{V_\text{CC}}}{\frac1{V_\text{CC}}}\cdot\frac{\text{d}\, V_\text{CC}}{V_\text{CC}-V_\text{LED}}-\frac{\frac1{V_\text{LED}}}{\frac1{V_\text{LED}}}\cdot\frac{\text{d}\, V_\text{LED}}{V_\text{CC}-V_\text{LED}}\\\\&=\frac{\frac{\text{d}\, V_\text{CC}}{V_\text{CC}}}{1-\frac{V_\text{LED}}{V_\text{CC}}}-\frac{\frac{\text{d}\, V_\text{LED}}{V_\text{LED}}}{\frac{V_\text{CC}}{V_\text{LED}}-1}\\\\&=\frac{\%\, V_\text{CC}}{1-\frac{V_\text{LED}}{V_\text{CC}}}-\frac{\%\, V_\text{LED}}{\frac{V_\text{CC}}{V_\text{LED}}-1}\end{align*}$$
This allows us to focus on \$\%\,V_\text{LED}\$, by taking the last term and its sign on the right side, or to focus on \$\%\,V_\text{CC}\$, by taking the first term and its sign on the left side. (Or, of course, to take both into account at the same time.)
