How do you find vertical, horizontal and oblique asymptotes for $( x + 5) /( 6 - x^3)$ ?

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Answer 1 · 2017-02-09T07:17:27

The vertical asymptote is $x=root(3)6$
The horizontal asymptote is $y=0$
No oblique asymptote

Let $f(x)=(x+5)/(6-x^3)$

As you cannot divide by $0$ , $x!=root(3)6$

The vertical asymptote is $x=root(3)6$

As the degree of the numerator is $<$ than the degree of the denominator, there is no oblique asymptote

$lim_(x->-oo)f(x)=lim_(x->-oo)x/(-x^3)=lim_(x->-oo)-1/x^2=0^-$

$lim_(x->+oo)f(x)=lim_(x->+oo)x/(-x^3)=lim_(x->+oo)-1/x^2=0^-$

The horizontal asymptote is $y=0$
graph{(y-(x+5)/(6-x^3))(y)=0 [-20.27, 20.27, -10.14, 10.14]}

How do you find vertical, horizontal and oblique asymptotes for ( x + 5) /( 6 - x^3)( x + 5) /( 6 - x^3)?