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?

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.
