So i have a function
$y=\frac{x^3+2x+9}{\sqrt{4x^2+3x+2}}$.
At wolfram alpha they say that it has a non linear asymptote $y=\frac{x^2}{2} - \frac{3x}{16}+ \frac{251}{256}$.
How do you predict this?
2026-04-14 05:15:54.1776143754
Predicting non linear asymptotes of a real valued function.
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$ y= \frac{x^3+2x+9}{2x} (1+\frac{3}{4x}+\frac{1}{2x^2})^{\frac{-1}{2}} $ ... binomially expand the bracket ... \begin{eqnarray*} \left(1+\frac{3}{4x}+\frac{1}{2x^2}\right)^{\frac{-1}{2}} &=&1+ \frac{-1}{2} \left(\frac{3}{4x}+\frac{1}{2x^2}\right) +\frac{\frac{-1}{2}(\frac{-1}{2}-1)}{2} \left(\frac{3}{4x}+\frac{1}{2x^2}\right)^2 +\cdots \\ &=&1+\frac{-3}{8x}+\frac{-5}{128x^2}+\cdots \end{eqnarray*} Now multiply this by $\frac{x^3+2x+9}{2x}$ ... we get $y=\frac{x^2}{2} - \frac{3x}{16}+ \frac{251}{256} +o(\frac{1}{x})$.