How do you find the asymptotes for f(x) = (x^2) / (x^2 + 1)?

1 Answer
Jan 22, 2016

cancel(EE) vertical asymptotes
y=1 horizontal asymptote x rarr +-oo

Explanation:

Given:

f(x)=g(h)/(h(x))

possible asymptotes are x=x_i with x_i values that zero the denominator h(x)=0.

Indeed, the existence condition of f(x) in RR is h(x)!=0.

Therefore,

h(x)=0 => x^2+1=0=> x^2=-1=>x_1,x_2 in CC

h(x) hasn't any zeros in RR

:.f(x): RR rarr RR, AA x in RR

We could looking for possible horizontal asymptotes:

y=L is a horizontal asymptotes when:

lim_(x rarr +-oo)f(x)=L

lim_(x rarr +-oo)f(x)=lim_(x rarr +-oo) x^2/(x^2+1)~~cancel(x^2)/cancel(x^2)=1

:.y=1 horizontal asymptote x rarr +-oo

graph{x^2/(x^2+1) [-6.244, 6.243, -3.12, 3.123]}