Peak of Poisson distribution I was experimenting with Poisson distribution on desmos and noticed that as $\lambda$ (the mean of distribution) gets larger, the x coordinate of distribution peak approaches $\lambda-0.5$. Why is it so?

The mode of the Poisson distribution occurs at the value $\text{Mode}(\lambda) = \lceil \lambda \rceil - 1$, whereas your proposed approximation is:
$$\widehat{\text{Mode}}(\lambda) \equiv \lambda - \tfrac{1}{2}.$$
Letting $u(\lambda) \equiv \lceil \lambda \rceil - \lambda$ denote the "upper remainder" of the number $\lambda$ we can write the difference between the true mode and your approximation as:
$$\begin{align}
\text{DIFF} \equiv
\text{Mode}(\lambda) - \widehat{\text{Mode}}(\lambda)
&= (\lceil \lambda \rceil - 1) - (\lambda-\tfrac{1}{2}) \\[6pt]
&= (\lceil \lambda \rceil - \lambda) -\tfrac{1}{2} \\[6pt]
&= u(\lambda) - \tfrac{1}{2}. \\[6pt]
\end{align}$$
It is simple to establish that $-\tfrac{1}{2} \leqslant \text{DIFF} < \tfrac{1}{2}$ so your approximation is near to the true mode.  As $\lambda \rightarrow \infty$ the relative error in the approximation converges to zero.
