Let $E/\mathbb{Q}$ be an elliptic curve, given by some minimal Weierstrass equation (say $Y^2 = X^3 + aX + b$ for some integer $a$ and $b$), and let $P$ be a rational point on $E$ which is not the infinite point. We assume that $P$ is not torsion.

The point $P$ can be written as $(\frac{x}{d^2}, \frac{y}{d^3})$ for some integers $x, y, d$ such that $d$ is prime to $x$ and $y$.

Now consider the sequence of points $P, 2P, 3P, \cdots$, we are led to three sequences $x_n, y_n, d_n$, such that $nP = (\frac{x_n}{d_n^2}, \frac{y_n}{d_n^3})$.

The question: does the limit $\lim_{n\rightarrow\infty} \frac{2\log(d_n)}{n^2}$ exist? Is it equal to the canonical height of $P$?

Same question for $x_n$ and $y_n$?

A related question:

Let us denote by $h(P)$ the natural height of $P$, i.e. $h(P) = \log(\max\{|x|, |d^2|\})$. Does the limit $\lim_{n\rightarrow\infty}\frac{h(nP)}{n^2}$ exist? (If it exists, then it of course is equal to the canonical height.)