confused about limit problem

[cotfw]

My book went over some examples of limit problems, but the exercises contain problems that are unlike the ones in the chapter, so I am confused.

LT x --> infinity (4x^2 + x - 1)/(3x^2 + 2x + 1)

I don't see an obvious way to factor either the numerator or the denominator. I thought maybe I should run the top and bottom through the quadratic formula but that didn't seem to help me at all. Do you know what I do?

 

[HallsofIvy]

The limit is as x goes to infinity, not any finite number, so factoring numerator and denominator won't help. Instead, seeing that the numerator and denominator are both polynomials of degree 2, divide both numerator and denominator by \(\displaystyle x^2\). That will make each a constant plus fractions with powers of x in the denominator: \(\displaystyle \frac{4+ \frac{1}{x}- \frac{1}{x^2}}{3+ \frac{2}{x}+ \frac{1}{x^2}}\). What happens to those fractions as x goes to infinity?

 

[cotfw]

The limit is as x goes to infinity, not any finite number, so factoring numerator and denominator won't help. Instead, seeing that the numerator and denominator are both polynomials of degree 2, divide both numerator and denominator by \(\displaystyle x^2\). That will make each a constant plus fractions with powers of x in the denominator: \(\displaystyle \frac{4+ \frac{1}{x}- \frac{1}{x^2}}{3+ \frac{2}{x}+ \frac{1}{x^2}}\). What happens to those fractions as x goes to infinity?

Ah that makes sense. I get 4/3 now, which the book says is right. Thanks for your help!