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]}