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# Examples of rational functions

Examples of rational functions are $$f(x) = \frac{ax+b}{dx+e},\quad g(x) = \frac{ax^2+bx+c}{dx+e}.$$ We assume that $a,d\ne 0$ and that $-e/d$ is not a zero of the numerator, so that the functions are in reduced form

• Domain. The functions are defined on $\R\setminus \{-e/d\}$
• Zero. Their zeros are the zeros of the numerator.
• Vertical asymptote: $x=-e/d$.
• Horizontal asymptote for $f$: $\displaystyle y = a/d$.
• Oblique asymptote for $g$. $$\displaystyle y = \alpha x + \beta,\quad \alpha = \frac{a}{d},\; \beta=\frac{bd-ae}{d^2}.$$

The oblique asymptote for $g$ is found from the computations ($x\to\infty$) \begin{align*} g(x) - \alpha x-\beta &= \frac{ ax^2+bx+c -\alpha d x^2 - (\alpha e + \beta d)x - \beta e}{ dx+e }\\ &= \frac{cd^2-bde+ae^2}{d^2(dx+e)}\to 0. \end{align*}

Rational functions: $f(x)=\frac{x}{x+1}$ and $g(x)=\frac{x^2-1}{x}$