What Are the Horizontal Asymptotes of These Functions?

[karush]

View attachment 9262

Ok this is what I posted on MeWe in MathQuiz

I'm pretty sure this can be solved just by a quick look at the powers

But probably I could of explained it better

I know the book says to take the Limit...

 

[HallsofIvy]

Yes, a horizontal asymptote involves the behavior of a function as x goes to plus or minus infinity so a limit is necessarily involved.

A "quick look" shows that A and B don't have asymptotes and that C has y= 0 as asymptote. Dividing both numerator and denominator of D by x gives $$\frac{5}{\frac{1}{x}- 1}$$ and taking the limit as x goes to plus or minusinfinity, y goes to -5. Dividing both numerator and denominator of E by $$x^2$$ gives $$\frac{20- \frac{1}{x}}{\frac{1}{x^2}+ 4}$$ and taking the limit as x goes to plus or minus infinity, y goes to 5.