Limits at Infinity Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Section 4.6 in our textbook, and particularly its “Limits at Infinity” subsection, discusses what it means for a function to have a limit as x approaches plus or minus infinity, and how to find such limits. This Canvas discussion is a place where you can start practicing some of the ideas in that reading, or asking questions about them, etc.

Calculations

Use algebra, limit laws, etc., plus probably a certain amount of intuition, to find the following limits. Places where you have to rely on intuition or informal reasoning about these limits would be good things to point out in this discussion.

1. \[\lim_{x \to \infty} \frac{n+1}{n}\]
2. \[\lim_{t \to \infty} \frac{t^2 - 1}{t}\]
3. \[\lim_{n \to \infty} \frac{2n^3 - 3n^2 + n}{6n^3}\]

Formal Definition

We haven’t looked a lot at formal definitions of limits in the past, but the ones for limits at infinity are relatively understandable (as formal definitions of limits go), and the textbook doesn’t, as far as I see, do a lot to justify some of the rules it uses. So see if you can use the formal definition of a limit at infinity (see the “Formal Definitions” part of the “Limits at Infinity” subsection) to prove that

This rule, and its generalization to limits of 1/xⁿ for any positive integer n, is a fundamental one for finding limits at infinity.