Question

Asimptota oblică la −∞ la graficul funcţiei f:R→R,
f(x)=x3−arctgx

Answer 1

$f:R\rightarrow R\ , \ \ f(x)=3x-arctgx$

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:
$lim_{x\rightarrow -\infty}(\frac{arctgx}{x})=\frac{M}{-\infty}=0$

$m=\lim_{x\rightarrow -\infty}\frac{x(3 - 0)}{x}=\boxed{3}$

[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:
$y=3x-\frac{\pi}{2}$