6.2.10 Asymptotes

Let us consider the function

${\displaystyle g:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ\setminus \{1\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & 1+\frac{1}{x-1}\hspace{0.5em}.\hfill \end{array}}$
Its mapping rule is a sum of a polynomial (of degree $0$) and a rational term. By finding the common denominator it is easy to transform $g(x)$ into a rational form in which the degree of the polynomial in the numerator equals the degree of the polynomial in the denominator:

${\displaystyle g(x)=1+\frac{1}{x-1}=\frac{x-1}{x-1}+\frac{1}{x-1}=\frac{x-1+1}{x-1}=\frac{x}{x-1}\hspace{0.5em}.}$
Thus, we can rewrite $g$ in the form

${\displaystyle g:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ\setminus \{1\}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{x}{x-1}\hspace{0.5em},\hfill \end{array}}$
and we now consider the corresponding graph.

Besides the pole and the singularity at $x=1$, we see that the value $y=1$ is of specific relevance. Obviously, this value is never taken by the function $g$. Thus, the range of $g$ is $W_g=ℝ\setminus \{1\}$. Instead, for "very large" values ($x$ tends to plus infinity) and "very small" values ($x$ tends to minus infinity) of the independent variable $x$, the function $g$ approaches its limiting value $1$ indefinitely without ever reaching it for any real number $x$.
This can be seen from the mapping rule $g(x)=1+\frac{1}{x-1}$ as follows. For "very large" ($50$, $100$, $1000$, etc.) or "very small" ($50$, $100$, $1000$, etc.) values of $x$, the rational part $\frac{1}{x-1}$ approaches $0$ since $x$ occurs in the denominator. In general, for these values of $x$, only the polynomial part $1$ of the mapping rule remains. This part can be described by a - in this case constant - function that is called an asymptote $g_{\mathrm{As}}$ of the function $g$:

${\displaystyle g_{\mathrm{As}}:\mathrm{\hspace{0.5em}\hspace{0.5em}}\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & 1\hspace{0.5em}.\hfill \end{array}}$
Since this is a constant function, it is also called a horizontal asymptote.