# How do you identify the oblique asymptote of ( 2x^3 + x^2) / (2x^2 - 3x + 3)?

Nov 4, 2015

See explanation.

#### Explanation:

You have to consider the extreme cases for asymptotes and these will vary according to the equation concerned. Usually it involves ${\lim}_{x - . \infty}$ but it may also involve${\lim}_{x \to \text{some number}}$. In your case I am assuming we are not looking at the behaviour near $x = 0$.

$\textcolor{b r o w n}{\text{Consider the numerator:}}$
$2 {x}^{3}$ grows much faster than ${x}^{2}$. Consequently $2 {x}^{3}$ 'wins' as the dominant factor as $x \to \infty$

$\textcolor{b r o w n}{\text{Consider the denominator:}}$
For the same reason as above, $2 {x}^{2}$ 'wins' as the dominant factor as $x \to \infty$.

$\textcolor{b r o w n}{\text{Put together:}}$
Thus as $x \to \infty$ we have ${\lim}_{x \to \infty} \frac{2 {x}^{3} + {x}^{2}}{2 {x}^{3} - 3 x + 3} = \frac{2 {x}^{3}}{2 {x}^{2}} = x$

Note that $x$ can be positive or negative.