ag.algebraic geometry – A question on linear projection of a smooth projective variety

Let $X$ be a smooth, projective $mathbb{C}$-variety of dimension $n$. Fix a closed point $x in X$ and an embedding of $X$ in $mathbb{P}^m$ for some integer $m$. For a given $d$, denote by $sigma_d : mathbb{P}^m to mathbb{P}^{N_d}$ the $d$-tuple embedding. My question is: for $d gg 0$, does there exist a linear subspace $L subset mathbb{P}^{N_d}$ of dimension $N_d-n-2$, not intersecting $sigma_d(X)$ such that for the linear projection from $L$ (sometimes called projection with centre $L$):
$$pi_L : sigma_d(X) to mathbb{P}^{n+1}$$ we have $pi_L^{-1}(pi_L(sigma_d(x)))=sigma_d(x)$ i.e., the preimage of $sigma_d(x)$ is only $sigma_d(x)$? Any hint/reference is most welcome.