Proper holomorphic embeddings with small limit sets

Let $X$ be a Stein manifold of dimension $n\ge 1$. Given a continuous positive increasing function $h$ on ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$ with ${\lim }_{t\to \mathrm{\infty }}h(t)=\mathrm{\infty }$, we construct a proper holomorphic embedding $f=(z,w):X\hookrightarrow {\mathbb{C}}^{n+1}\times {\mathbb{C}}^n$ satisfying $\mathrm{\mid }w(x)\mathrm{\mid }<h(\mathrm{\mid }z(x)\mathrm{\mid })$ for all $x\in X$. In particular, $f$ may be chosen such that its limit set at infinity is a linearly embedded copy of ${\mathbb{C}\mathbb{P}}^n$ in ${\mathbb{C}\mathbb{P}}^{2n}$.