Question

How do you prove there is no Vertical Asymptote in (3x^4 + 5x +1)/(1+x^2) using limits?

The rule is:

Lim x--> b+ f(x) and Lim x--> b- f(x) = + infinity on one side and - infinity on the other
Then Lim x--> b DNE, so there is a vertical asymptote

What is b in this case? What should x be approaching?

Answer 1

the way you get vertical asymptotes is to divide by 0
in this case the denominator 1+x^2 is always positive and 1 or greater, so you will never divide by 0.
if you had 1-x^2 then x=1 or x=-1 would cause a divide by 0... and 1 or -1 would be "b" in your limits.

But I am not sure what they want as proof there is no asymptote for this problem

Answer 2

Thank you I understand it!