<s>[INST] How to find the oblique asymptote of root of a function? In a test example I'm solving, the question asks to find the oblique asymptote of the following function:
$f(x) = \sqrt{4x^2+x+6}$ 
$x$ at $+\infty$
We have only learned how to do so with rational functions. Is there any general way of finding the oblique asymptote that works with any kind of function? Perhaps using limits?
 [/INST] By completing the square we get $f(x) = \sqrt{ (2x + 1/4)^2 + 95/16}$. 
This means (after squaring both sides and taking $(2x + 1/4)^2$ to the left hand side and factoring) that $$( f(x) - (2x + 1/4) ) ( f(x) + (2x + 1/4) ) = 95/16$$ and hence $$f(x) - (2x + 1/4)  =  \frac{95/16}{f(x) + (2x + 1/4)}.$$
But $f(x) + (2x + 1/4) \rightarrow \infty$ as $x \rightarrow \infty$. This implies that 
$$f(x) - (2x + 1/4) \rightarrow 0$$  as $x \rightarrow \infty$.
Remark: Likewise  $$f(x) + (2x + 1/4)  =  \frac{95/16}{f(x) - (2x + 1/4) }$$ and hence as  $x \rightarrow -\infty, f(x) - (2x + 1/4) \rightarrow \infty$ and hence
$$\frac {95/16}{f(x) - (2x + 1/4)} \rightarrow 0.$$ This implies that $f(x) + (2x + 1/4) \rightarrow 0$ and thus $y = -(2x + 1/4)$  is another asymptote.
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