Infinite Limits

Anything You Want to Talk About?

Especially 1-sided limits or other things from days you didn’t get to come to class in person?

What’s the idea about 1-sided limits existing from one side but not the other, or something...? The most important existence issue involving 1-sided limits is the connection between them and the existence of 2-sided limits. Usually 1-sided limits either both exist or both don’t exist, and the question is “if they both exist, are they equal to each other?” If they are, then the 2-sided limit also exists and is equal to the 1-sided ones; if the 1-sided limits are unequal then the 2-sided limit does not exist. One situation in which a limit from one side might exist and the limit from the other side not is if the non-existent limit is infinite. Here’s a graph the demonstrates all of these possibilities (from right to left, equal 1-sided limits, unequal 1-sided limits, and only 1 1-sided limit existing):

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Infinite Limits

From “Infinite Limits” in section 2.2 of the book, and the infinite limits discussion.

Key Ideas

The “Infinite Limits from Positive Integers” rule from the book is basically a limit law for finding infinite limits; know how to use it. See “The Idea” below.

You can use other limit laws to decompose an expression that isn’t immediately in a form the “Infinite Limits from Positive Integers” law applies to into a sum, product, etc. of simpler limits, some of which the law does apply to. See “Using Limit Laws” below.

The Idea

Use the “Infinite Limits from Positive Integers” rule from the textbook to find the following limits.

The general strategy for using this rule is something like this:

1. Identify the constant a from the limit and the expression, and be sure they are the same.
2. Identify the exponent n in the expression and decide whether it’s odd or even.
3. If the exponent is odd, the 1-sided limits are +∞ and -∞, so the 2-sided limit does not exist.
4. If the exponent is even, the 2-sided limit is +∞, and so both 1-sided limits are too.

1/x³:

Here a is 0 and n is odd, so we can read 1-sided limits from the rule, and the 2-sided limit doesn’t exist:

1/(x-1)⁴:

Now a is 1 and n is even, implying that all the limits are +∞:

Note that if “a” in the expression whose limit you’re finding, and “a” from the limit itself don’t match, you can’t use this infinite limit law, but on the other hand you can probably find the limit by just plugging “a” from the limit into the expression:

Using Limit Laws

Consider finding

The infinite limit law doesn’t immediately seem to apply. But this can be written as a product in which the law would apply to one of the terms, and the other one could be evaluated by older limit laws:

The details of doing this work like so:

Beware of arithmetic involving infinite values, infinity is not a number. In general, it’s safe to assume that multiplying by infinity produces an infinite result, and adding a finite and infinite value produces an infinite result, but subtracting infinities from each other or dividing them is undefined. For example, here is a valid use of the idea of using limit laws to decompose an expression into parts to which the infinite limit laws can be applied ultimately failing because it leads to ∞ - ∞ (but we know how to find the limit in question by other means):

Next

Continuity, i.e., what does the phrase “continuous function,” which many of you have probably heard, really mean.

Read “Continuity at a Point” and “Types of Discontinuities” in section 2.4 of the textbook by class time Thursday.

Also participate in this discussion by class time Thursday.