Long division - Rational functions

We have seen in the previous section that a curve like $y = 1 + \frac{2}{x+3}$ has asymptotes $x=-3$ and $y=1.$

Now let us look at a curve described by an improper fraction $y = \frac{6x-1}{3x+2}$ . The horizontal asymptote $x=-\frac{2}{3}$ can be deduced easily, but the vertical asymptote is harder to figure out as both $6x-1$ and $3x+2$ tend to infinity as $x \to \infty.$

We will tackle this, as well as more complicated improper fractions like $y = \frac{2x^2+7x-3}{x+3}$ , by long division.

Examples

{2}

{3x+2}

{)}

{6x}

{-}

{1}

{-(}

{6x}

{-}

{4)}

{3}

{2x}

{+}

{2}

{x+3}

{)}

{2x^2}

{+}

{7x}

{-}

{3}

{-(}

{2x^2}

{+}

{6x)}

{x}

{-}

{3}

{-(}

{x}

{+}

{3)}

{-}

{6}

Asymptotes

Since $\displaystyle {y = \frac{6x-1}{3x+2}} \allowbreak {= 2 + \frac{3}{3x+2}}$ , we can see clearly that the horizontal asymptote is $y=2$ .

We will tackle the result of the second example $\displaystyle {y = \frac{2x^2+7x-3}{x+3}} \allowbreak {= 2x+2 - \frac{6}{x+3}}$ in section 3.1.4.

In the next section, however, we will first look at more examples of the first type.