Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:

1. Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without further conditions $f$ cannot be injective (for e.g. take $f(x) = x \sin(x)$ for $n=1$). I want to know if any generic conditions are known such that $f$ is eventually injective (meaning that there exists $b > a$ s.t $f$ is injective on $(b, \infty)$.

2. Suppose $f(x)$ converges as $x \rightarrow \infty$, and $(\partial^m f / \partial x^m)(x) \rightarrow 0$ as $x \rightarrow \infty$, for all $m \geq 1$. Again without further conditions $f$ is not eventually injective (for e.g. $f(x) = e^{-x} \sin (x)$ shows it is not true generically). So again my question is whether there are some simple conditions known that makes $f$ eventually injective.

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:

1. Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without further conditions $f$ can be not injective (for e.g. take $f(x) = x + 2 \sin(x)$ for $n=1$). I want to know if any generic conditions are known such that $f$ is eventually injective (meaning that there exists $b > a$ s.t $f$ is injective on $(b, \infty)$.

2. Suppose $f(x)$ converges as $x \rightarrow \infty$, and $(\partial^m f / \partial x^m)(x) \rightarrow 0$ as $x \rightarrow \infty$, for all $m \geq 1$. Again without further conditions $f$ is not eventually injective (for e.g. $f(x) = e^{-x} \sin (x)$ shows it is not true generically). So again my question is whether there are some simple conditions known that makes $f$ eventually injective.

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:

1. Suppose $||f(x)|| \rightarrow \infty$ diverges as $x \rightarrow \infty$. I know without further conditions $f$ cannot be injective (for e.g. take $f(x) = x \sin(x)$ for $n=1$). I want to know if any generic conditions are known such that $f$ is eventually injective (meaning that there exists $b > a$ s.t $f$ is injective on $(b, \infty)$.

2. Suppose $f(x)$ converges as $x \rightarrow \infty$, and $(\partial^m f / \partial x^m)(x) \rightarrow 0$ as $x \rightarrow \infty$, for all $m \geq 1$. Again without further conditions $f$ is not eventually injective (for e.g. $f(x) = e^{-x} \sin (x)$ shows it is not true generically). So again my question is whether there are some simple conditions known that makes $f$ eventually injective.

Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:

1. Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without further conditions $f$ cannot be injective (for e.g. take $f(x) = x \sin(x)$ for $n=1$). I want to know if any generic conditions are known such that $f$ is eventually injective (meaning that there exists $b > a$ s.t $f$ is injective on $(b, \infty)$.

2. Suppose $f(x)$ converges as $x \rightarrow \infty$, and $(\partial^m f / \partial x^m)(x) \rightarrow 0$ as $x \rightarrow \infty$, for all $m \geq 1$. Again without further conditions $f$ is not eventually injective (for e.g. $f(x) = e^{-x} \sin (x)$ shows it is not true generically). So again my question is whether there are some simple conditions known that makes $f$ eventually injective.