Question

What is the oblique asymptote of the function f(x) x^2+x-2/x+1

Answer 1

$f(x)=\displaystyle\frac{x^2+x-2}{x+1}=\frac{x(x+1)-2}{x+1}=x-\frac{2}{x+1}$

$\displaystyle\lim_{x\to\infty}\left\{f(x)-x \right\}=\lim_{x\to\infty}\frac{2}{x+1}=\lim_{x\to\infty}\frac{\displaystyle\frac{2}{x}}{1+\displaystyle\frac{1}{x}} =0$

Consequently, the limit of $f(x)$ as x approaches infinity is $x$.

In other words, $f(x)$ approaches the line y=x, 

so oblique asymptote is y=x.

I'm Japanese, if you find some mistakes in my English, please let me know.