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I thought it is enough to consider oblique asymptote for the equation whose the numerator's degree of polynomial is higher than the denominator's in order to draw a graph.

But the equation $x + arctan(x)$ is also needed for the graph in my notebook. What is the condition that we should get a oblique asymptote of a equation?

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    Only $\displaystyle \lim_{x\to\infty}\frac{y}{x}=m>0$, For yours$\displaystyle \lim_{x\to\infty}\frac{x+\arctan x}{x}=1>0$. – 2017-01-10
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    Your first paragraph describes (not quite accurately, though) the recipe for finding asymptotes for **rational functions** only. (Recall that a rational function is the ratio of two polynomials.) But there are many more other functions out there, and some of those other types of functions may have asymptotes as well. The one in your question, $f(x)=x+\arctan(x)$, is such an example. – 2017-01-10

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Since $\lim_{x\to +\infty}\arctan x=\pi/2$ (resp. $\lim_{x\to -\infty}\arctan x=-\pi/2$), it is obvious the oblique asymptotes for the representative curve of this function are $$y=x+\frac\pi2, \quad(\text{resp.} \enspace y=x-\frac\pi2).$$

General method:

Suppose a function $f$ is such that $\lim_{x\to\infty}f(x)=\infty$. One first has to compute $\displaystyle\lim_{x\to\infty}\frac{f(x)}x=\ell$. If such a limit exists, it is said that the graph of $f$ has an asymptotic direction with slope $\ell$.

  • If $\ell=\infty$, we actually have a vertical parabolic branch;
  • If $\ell=0$, we have a horizontal parabolic branch;
  • If $\ell\neq 0,\infty$, one has to determine $\;\lim_{x\to\infty}(f(x)-\ell x)=m$.
    Supposing this limit exists, if $m=\infty$, the graph has a parabolic branch in the direction with slope $\ell$. If $m<\infty$, the graph has an oblique asymptote with equation $\;y=\ell x+m$.

When possible, it is often shorter to make a change of variable $t=1/x$, and use Taylor's expansion of $f(1/t)$ near $t=0$, with two terms. In favourable cases, the expansion will look like $$f(1/t)= \frac{\ell}t+m+o(1)=\ell x+m+o(1),$$ thus proving there is an oblique asymptote, with equation $y=\ell x+m$.