How to Evaluate Limits at Infinity for Rational Functions in Calculus?

Question

How do you find the limit of a function as it approaches infinity, specifically when dealing with indeterminate forms? For example, how would you solve \lim_{x \to \infty} \frac{3x^2 + 5x - 2}{2x^2 - 4x + 7}?

This question allows you to discuss L’Hopital’s Rule, the behavior of functions as they approach infinity, and how to simplify complex rational functions. Additionally, you could expand on alternative methods like factoring out the highest power of x in the numerator and the denominator.

Answer 1

you solve \lim_{x \to \infty} \frac{3x^2 + 5x - 2}{2x^2 - 4x + 7}?

lim (x -> inf) of (3x^2 + 5x - 2) / (2x^2 - 4x + 7)

It is infinite over infinite.

1. Without L'H:

I will use as x approaches to infinite, lim 1/(x^n) = 0, where n > 0

I will divide all the terms by x^2

lim (x -> inf) of (3 + 5/x - 2/x^2) / (2 - 4/x + 7/x^2), all the terms with over x or x^2 go to 0.

so, limit is equal to 3/2

1. With L'H

lim (x -> inf) of (3x^2 + 5x - 2) / (2x^2 - 4x + 7), we take a derivative numerator and denominator separately

= lim (x -> inf) of (6x + 5) / (4x - 4), still inf / inf, we take another derivative

= lim (x -> inf) of 6 / 4 = 6/4 = 3 / 2