<s>[INST] On understanding the use of binomial theorem to find asymptotes of a real valued function. I have the function 
$$y=\frac{x^3+2x+9}{\sqrt{4x^2+3x+2}}$$
Wolfram Alpha says that it has a non linear asymptote at
$y=\frac{x^2}{2} - \frac{3x}{16}+ \frac{251}{256}$. 
I have been told to expand using the binomial theorem to obtain the asymptote at predicting-non-linear-asymptotes-of-a-real-valued-function ; however, I do not understand this. How does expanding binomially give me the asymptote ?
 [/INST] hint
Let $$f (x)=(x^3+2x+9)(4x^2+3x+2)^\frac{-1}{2}$$
$$g (x)=f (1/x)=$$
$$(1+\frac {9x}{2}+\frac {1}{2x^2})(1+\frac {3x}{4}+\frac {x^2}{2})^\frac {-1}{2} $$
expand $g (x) $ near $x=0$ using binomial formula.
$$(1+\frac {3x}{4}+\frac {x^2}{2})^\frac {-1}{2} $$
$$=1-\frac {3x}{8}-\frac {x^2}{4}+\frac {27x^2}{112}+x^2\epsilon (x)$$
You will get $f (x)=g (1/x) $ and the asymptote.
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