How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=x/(x^2+4)#?

1 Answer
Jan 28, 2017

Answer:

The horizontal asymptote is #y=0#
No vertical asymptote
No oblique asymptote

Explanation:

The domain of #f(x)# is #D_f(x)=RR#

The denominator is #>0 AA x in RR#

There are no vertical asymptotes.

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^2=lim_(x->-oo)1/x=0^-#

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

The horizontal asymptote is #y=0#

graph{(y-x/(x^2+4))(y)=0 [-5.55, 5.55, -2.773, 2.776]}