I'm looking for a way of finding an oblique asymptote of (on infinity):
\begin{equation} \sqrt{1 + x^2 + \sqrt{(1 + x^2)^2 - 2 x^2 \cos^2{\theta}}} \end{equation}
I know that the asymptote is $\sqrt{2}x$.
I'm trying to find it simply by finding limit on the infinity:
\begin{align} \lim\limits_{x\to\infty}\sqrt{1 + x^2 + \sqrt{(1 + x^2)^2 - 2 x^2 \cos^2{\theta}}} & = |\text{Omitting limit sign for a while}| = \\ = |\text{Take }\theta\text{ be equal to it max value}| &= \sqrt{1 + x^2 + \sqrt{1 + 2 x^2 + x^4 - 2 x^2}} = \\ = |\text{Omitting 1}| & = \sqrt{x^2 + \sqrt{x^4}} = \\ = |\text{Simplifying}| & = \sqrt{2} x \end{align}
It this way of finding this asymptote is correct? I'm mostly confused with the step when I'm omit the $1$ comparing to $x^n$.