Question

find the horizontal asymptotes of: (sqrt(9x^2 + 2x + 5) - 4) / (x - 1)

Answer 1

if you factor out an x^2 from the radicand, and then apply the sqrt you will get an |x| in the numerator\[f(x)=\frac{|x| \sqrt{9 + 2/x +5/x^2}}{x-1}\]Now to the right\[\lim_{x \rightarrow +\infty}f(x)=\lim_{x \rightarrow +\infty}\frac{|x| \sqrt{9}}{x}=\sqrt{9}=3\]Out to the left\[\lim_{x \rightarrow -\infty}f(x)=\lim_{x \rightarrow -\infty}\frac{|x|\sqrt{9}}{x}=-3\]Thus the horontal asymptotes are y=-3 (to the left) and y=3 (to the right).

See attached-- a little hard to read as HA don't graph well to the eye.

Answer 2

thankyou!

Answer 3

np :})

Answer 4

what about the vertical asymptotes?