[tex]f:R\rightarrow R\ , \ \ f(x)=3x-arctgx[/tex]
Asimptota oblica este o dreapta y = mx + n, unde:
[tex]m=\lim_{x\rightarrow -\infty}\frac{f(x)}{x}=\lim_{x\rightarrow -\infty}\frac{3x-arctg(x)}{x}=\\\\
\lim_{x\rightarrow -\infty}\frac{x(3 - \frac{arctg(x)}{x})}{x}[/tex]
Stim ca functia arctg(x) este marginita (imaginea este (-π/2, π/2)). Asadar:
[tex]lim_{x\rightarrow -\infty}(\frac{arctgx}{x})=\frac{M}{-\infty}=0[/tex]
[tex]m=\lim_{x\rightarrow -\infty}\frac{x(3 - 0)}{x}=\boxed{3}[/tex]
[tex]n=lim_{x\rightarrow -\infty}(f(x)-mx)=lim_{x\rightarrow -\infty}(3x+arctgx-3x)=\\\\
lim_{x\rightarrow -\infty}(arctgx)=\boxed{-\frac{\pi}{2}}[/tex]
Asimptota oblica la minus infinit:
[tex]y=3x-\frac{\pi}{2}[/tex]
