For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$, the limit of $u_n/v_n$ as $n\to\infty$, when exists, is called the Apéry limit. We give a construction that realises certain quotients of $L$-values of elliptic curves as Apéry limits.