On the one hand, given an open Riemann surface $\mathcal{N}$ and a real number $\theta \in ]0,\pi/4[$, we construct a conformal minimal immersion $X = (X_1, X_2, X_3): \mathcal{N} \to \mathbb{R}^3$ such that $X_3 + \tan \theta |X_1|: \mathcal{N} \to \mathbb{R}$ is positive and proper. Furthermore, $X$ can be chosen with arbitrarily prescribed flux map. This construction if related with a problem posed by Shoen and Yau. On the other hand, we produce properly immersed hyperbolic minimal surfaces with empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph. This construction can be linked to a conjecture by Meeks.
The main tool used in the construction of the above examples is an approximation theorem by minimal surfaces. We will also remark some other applications of this result.