How do you identify all asymptotes or holes for $f (x) = \frac{x^{2} - 2}{x}$ $f(x)=(x^2-2)/x$ ?

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Answer 1 · 2017-01-18T07:58:11

The vertical asymptote is $x = 0$ $x=0$
The slant asymptote is $y = x$ $y=x$
No holes and no horizontal asymptote.

The domain of $f (x)$ $f(x)$ is $D_f(x)=RR-{0}$

As we cannot divide by $0$ , $x!=0$

The vertical asymptote is $x=0$

We can rewrite $f(x)$ as

$f(x)=(x^2-2)/x=x-2/x$

Therefore,

$lim_(x->-oo)(f(x)-x)=lim_(x->-oo)-2/x=0^+$

$lim_(x->+oo)(f(x)-x)=lim_(x->+oo)-2/x=0^-$

The slant asymptote is $y=x$

graph{(y-(x^2-2)/x)(y-x)=0 [-11.25, 11.25, -5.625, 5.62]}

How do you identify all asymptotes or holes for f(x)=(x^2-2)/xf(x)=x2−2xf(x)=(x^2-2)/x?