How do you find the vertical, horizontal or slant asymptotes for x/sqrt(4x^2-1)x4x21?

1 Answer
Jul 2, 2016

vertical at x = pm 1/2x=±12

horizontal asymptotes are lim_{x to pm oo} f(x) = pm 1/2

Explanation:

x/sqrt(4x^2-1)

for vertical we look at when the numerator is zero

ie sqrt(4x^2-1) = 0
4x^2=1
x = pm 1/2

because sqrt(4x^2-1) >0 forall x notin (- 1/2,1/2)

the limit follows the sign of the numerator so

lim_{x to -(1/2) ^ -} = - oo

lim_{x to (1/2) ^+} = + oo

for horizontal asymptote we look at

lim_{x to pm oo} x/sqrt(4x^2-1)

lim_{x to pm oo} sgn(x) 1/sqrt(4-1/x^2)

lim_{x to pm oo} 1/2 sgn(x) = pm 1/2