How do you find all the asymptotes for $\frac{3 x^{2} + 2 x - 3}{x - 1}$ $(3x^2 + 2x - 3)/(x-1 )$ ?

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How do you find all the asymptotes for $\frac{3 x^{2} + 2 x - 3}{x - 1}$ $(3x^2 + 2x - 3)/(x-1 )$ ?

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Answer 1 · 2015-11-09T13:41:44

graph{(3x^2+2x-3)/(x-1) [-35.6, 41.33, -13.23, 25.24]}

$x = 1$ $x=1$ is a vertical asymptote.
$y = 3 x + 5$ $y=3x+5$ is an oblique asymptote.

Explanation:

First simplify the expression by performing long division.

$\frac{3 x^{2} + 2 x - 3}{x - 1} \equiv 3 x + 5 + \frac{2}{x - 1}$ $frac{3x^2+2x-3}{x-1}-=3x+5+frac{2}{x-1}$

As $x \to \pm \infty$ $x->+-oo$ , $\frac{3 x^{2} + 2 x - 3}{x - 1} \to 3 x + 5$ $frac{3x^2+2x-3}{x-1}->3x+5$

$y = 3 x + 5$ $y=3x+5$ is an oblique asymptote.

As $x \to 1^{+}$ $x->1^+$ , $\frac{3 x^{2} + 2 x - 3}{x - 1} \to + \infty$ $frac{3x^2+2x-3}{x-1}->+oo$
As $x \to 1^{-}$ $x->1^-$ , $\frac{3 x^{2} + 2 x - 3}{x - 1} \to - \infty$ $frac{3x^2+2x-3}{x-1}->-oo$

$x = 1$ $x=1$ is a vertical asymptote.

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How do you find all the asymptotes for $\frac{3 x^{2} + 2 x - 3}{x - 1}$ $(3x^2 + 2x - 3)/(x-1 )$ ?

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Explanation:

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