How do you find the asymptotes for #g(x)=x/(root4(x^4+2))#?

1 Answer
Feb 15, 2016

Answer:

Evaluate the limits of #g(x)# as #x->+oo# and #x->-oo# to find horizontal asymptotes #y=1# and #y=-1#.

Explanation:

#g(x) = x/root(4)(x^4+2) = x/abs(x) abs(x)/root(4)(x^4+2) = x/abs(x) 1/root(4)(1+2/x^4)#

So #lim_(x->+oo) g(x) = 1# and #lim_(x->-oo) g(x) = -1#

So #g(x)# has horizontal asymptotes #y=1# and #y=-1#

#x^4+2 >= 2 > 0# for any Real number #x#

Hence the denominator of #g(x)# is always non-zero and #g(x)# has no vertical asymptotes.

graph{x/root(4)(x^4+2) [-5.55, 5.55, -2.775, 2.774]}