Question

f functie f:/(-1,+infinit)->rf(x)=(x^2-2)(/x+1) determinati asimptotele graficului functiei

Answer 1

[tex]i)\text{Asimptota orizontala:}\\ \displaystyle\limit\lim_{x\to \infty} f(x)=\displaystyle\limit\lim_{x\to \infty} \dfrac{x^2-2}{x+1} =\infty \\ \text{Deci nu exista asimptote orizontale.}\\ \\ ii)\text{Asimpote verticale:}\\ \displaystyle\limit\lim_{x\searrow -1} f(x)= \displaystyle\limit\lim_{x\searrow -1} \dfrac{x^2-2}{x+1}= \dfrac{-1}{0^{+}}=\infty\\ y=-1 \text{ asimptota verticala.}\\ [/tex]

[tex] iii)\text{Asimptote oblice:}\\ m=\displaystyle\limit\lim_{x\to \infty} \dfrac{f(x)}{x}=\displaystyle\limit\lim_{x\to \infty} \dfrac{x^2-2}{x^2+x}= 1 \Rightarrow m=1\\ n=\displaystyle\limit\lim_{x\to \infty}(f(x)-x)=\displaystyle\limit\lim_{x\to \infty} \left(\dfrac{x^2-2}{x+1}-x\right)=\displaystyle\limit\lim_{x\to \infty} \dfrac{-x-2}{x+1}=-1\Rightarrow \\ \Rightarrow n=-1\\ y=x-1\text{ asimptota oblica} [/tex]