Lets be $f: \mathbb{R}^2 \longrightarrow \mathbb{R}$ and $g:\mathbb{R} \longrightarrow \mathbb{R}$. If $\epsilon(x)=\alpha x+\beta$ is a linear approximation of $g$ at $x_0 \in \overline{\mathbb{R}}$ (tangent or asymptote), what are the conditions of $f,g, \epsilon$ that allow me to whrite: $\lim \limits_{x \to x_0} f(x, g(x))=\lim \limits_{x \to x_0} f(x, \epsilon(x))$ ?
I want something more general than continuity of $f$.