Problem 599
Find all asymptotes of y=2x+ arctan dfrac x2
Find all asymptotes of $y=2x+\arctan\dfrac x2$.
Explanation of Method
A bounded correction term to a linear function still produces two distinct parallel asymptotes.
Step 1: No vertical asymptote
The function is continuous on all of $\mathbb R$.
Step 2: Oblique asymptote — slope
\[ k = \lim_{x\to\pm\infty}\!\left(2+\frac{\arctan(x/2)}{x}\right) = 2. \tag{1} \]
Step 3: Oblique asymptotes — intercepts
\[ b = \lim_{x\to\pm\infty}\arctan\frac x2 = \pm\frac\pi2. \tag{2} \]
Step 4: Verification
\[x=1000\text{: } y-(2x+\pi/2)\approx-2\times10^{-3}\to0;\qquad x=-1000\text{: } y-(2x-\pi/2)\approx2\times10^{-3}\to0\ \checkmark\]
Result
Final answer:
\[\boxed{\,y = 2x+\frac\pi2\ (x\to+\infty); \qquad y = 2x-\frac\pi2\ (x\to-\infty)\,}\]
Two distinct parallel oblique asymptotes, both of slope $2$.