Limits at Infinity

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Anything You Want to Talk About?

Do students have to grade problem sets in order? No, you can grade them in any order you want (as long as they all get graded by the end of the semester).

Misc

Colloquium

Tomorrow afternoon at 4:00, via Zoom .

“Idempotents a la Mod”

By Thomas Q. Sibley, College of St. Benedict and St. John’s University.

Exploring what happens with the simple equation x² = x when all numbers are represented by their remainders after division by some integer n (numbers “mod n”).

SI

Tuesday 6:30 - 8:00, watch for the link around 5:30.

Limits at Infinity

From “Limits at Infinity” in section 4.6 of the textbook and this discussion of limits at infinity.

Calculations

Find the limits as x approaches infinity (positive infinity, the right end of the x axis) of...

(x + 1) / x: start by using algebra and limit laws to break the limit into simpler ones, then use intuition that 1/x gets ever smaller as x gets ever bigger to conclude that its limit is 0:

Moral: use algebra and limit laws to simplify limits at infinity.

( 2x³ - 3x² + x ) / 6x³: Use more or less the same strategy as before, there’s just more algebra to do:

This and the previous example suggest a goal when simplifying: try to get the thing whose limit you want into the form of a constant plus or minus constants divided by x or powers of x. The former is its own limit, and the rest all go to 0 as x approaches infinity.

But that’s not always possible. Consider the limit as x goes to infinity of (x² - 1) / x:

At this point, there’s no way this is going to become a constant plus or minus fractions with x in the denominator. But split it up via the difference limit law anyhow, and then take the individual limits:

Moral: Sometimes limits at infinity are themselves infinite.

Finally, consider the limit at infinity of ( x² + 2x - 1 ) / (2x² + 1): This doesn’t give a nice decomposition into fractions that simplify into things whose limits are easy to take. It’s tempting to consider using the quotient limit law on it, but that law doesn’t help when either or both of the numerator and denominator are undefined (including infinite). But if you divide the numerator and the denominator both by x² (equivalently, multiply the whole fraction by (1/x²) / (1/x²)), something interesting does happen:

Now you have a form you can use the quotient law on, and both numerator and denominator are in the “constant plus constants over powers of x” form that we have seen before:

This illustrates a trick for limits at infinity of complicated rational functions (i.e., fractions whose numerators and denominators are polynomials): divide the numerator and denominator by the highest power in the denominator and then use limit laws.

Next

We used the idea that the limit as x approaches infinity of 1/x is 0 a lot above. The idea makes intuitive sense, but it would be nice to prove it.

So continue the conversation about limits at infinity, especially the formal definition.

Review “Formal Definitions” in the “Limits at Infinity” reading and keep contributing to the discussion of limits at infinity, especially the part about formal definitions and proofs, by class time Wednesday.

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