Approximating Functions to Order $k$

Let $f(x)$ be defined on a domain $D$, and let $g(x)$ be a function different from $f(x)$. If at some point $x = \xi$ we have

where $k$ is a positive integer and $f^{(k)}$ denotes the $k$-th derivative, we call $g(x)$ an order-$k$ approximation of $f(x)$ at $x = \xi$. For example, the tangent line $y = g(x)$ to $f(x)$ at $x = x_0$ is an order-$1$ approximation of $f(x)$ there.

1. Find an order-$2$ approximation of $f(x) = x^3$ at $x = 1$. 2. Let $f(x) = \ln x$ and $g(x) = ax + \dfrac{b}{x} + c$ with $a, b, c \in \mathbb{R}$. Can $g(x)$ be an order-$2$ approximation of $f(x)$ at $x = 1$? 3. Let $f(x) = \ln(x+1)$ and $g(x) = \dfrac{ax}{x+b}$ with $a, b \in \mathbb{R}$. Can $g(x)$ be an order-$2$ approximation of $f(x)$ at $x = 0$?