We introduce the notation and formal algebraic definitions for limits.
You may have noticed at this point that it can get cumbersome and tiring to keep saying the phrase “the limit of as approaches ” and mathematicians are inherently lazy, so we have defined a notation that encapsulates this entire phrase, which we present next.
This notation breaks down the limit information we need into neat little blocks, which we will pull apart and parse here.
The notation above is pretty dense and something that we will be using a lot. Take a few moments to really digest what the notation means. What would it mean to change the value? How about the value, or ? Make sure you understand the notation well before moving forward as we will be adding a couple more notational bits which are subtle, but very important.
Thus far we have used tables and graphs to determine limits, and we’ve discussed the possible up and downsides of both. Our next goal is to formalize the analytical idea of the limit; which requires the introduction of the one-sided limit. Recall that it can be helpful to break the region near our point of interest into the (nearby) region to the left of and the (nearby) region to the right of . We formalize the notation for these left and right limits next.
Similarly, for the function , is the limit from the left as approaches , written if the value of is as close as one wishes to for all sufficiently close to .
Recall that a limit exists at and equals if; for values sufficiently close to , gets as close as we want to . But notice this is equivalent to saying that both the left and right limits are equal to the same value; namely to . In other words, a limit exists if, and only if, both one-sided limits exist, and have the same value! We formalize this idea in the following theorem.
In this case, is equal to the common value of the two one sided limits.
Often in literature (like textbooks or notes) you will see the above theorem written as the following:
Notice that this is actually the same thing, but this definition is a little less clear. Specificallyand
can only equal L (and each other) if they exist. So this definition is hiding the fact that you need to check if the one sided limits exist in the first place by burying it as an assumption into the equality. You can use either definition (they are equivalent after all), but remember that this second definition has more going on than first meets the eye!
You should start by...
Optional Content
If you are interested in how mathematicians actually prove a limit’s value rigorously you can watch the following video on how mathematicians tackle the problem of testing all nearby points simultaneously. [URL to YouTube video, but not an embedded video.]
