SUNY Geneseo Department of Mathematics
Wednesday, November 11
Math 221 02
Fall 2020
Prof. Doug Baldwin
(No.)
There’s a session tonight, 5:00 - 6:30.
The next one will be Sunday, 6:00 - 7:30.
Based on “Formal Definitions” in the “Limits at Infinity” reading, and the discussion of limits at infinity, especially the part about formal definitions and proofs..
What is the formal definition of a limit at infinity, and what does it say in English?
Read the definition (at the bottom of page 413 in the PDF textbook), using the ideas we’ve talked about earlier for reading math (e.g., read slowly, stop often to check understanding or paraphrase ideas, try imagining or actually drawing pictures of situations).
The first 10 words are “We say a function f has a limit at infinity,...” Even this short snippet has a couple of ideas in it worth stopping to notice. First, the boldface “limit at infinity” indicates that that phrase is a technical term being defined. On the one hand, this entitles you to simply treat “limit at infinity” as a label for the ideas that are about to come up, without trying to impose your own understanding of “limit,” “infinity,” etc. on it. On the other hand, you might want to think a little about what you understand limits and infinity to be and be prepared to see if that understanding fits into the rest of the definition (but if it doesn’t, go with the definition, not what you believed before). Second, you have been introduced to one of the variables that will be used later, namely f, and told what it represents (a function). This is information worth remembering, or writing down in notes if you’re taking them, for later in the definition.
The next few words introduce another variable, L, and tell you that it stands for a real number. Again, that’s information to remember or write down.
Then the definition introduces 2 more variables, ε and N. It doesn’t say directly what they represent, but it implies that they are real numbers because both are greater than 0. So you know a little bit about their values. My reading process probably creates a little mental picture of ε and N at this point, something like a number line with ε and N on the positive side...
Finally, the definition ties 3 of these variables together with the inequality | f(x) - L | < ε. This takes some interpretation, interpretation that’s not necessarily obvious if you haven’t seen it before. f(x) - L is the distance between the value of f(x) and the value L; taking the absolute value makes it purely a distance, with no positive or negative sign to indicate whether f(x) is larger than L or vice versa. So the inequality is saying the f(x) is less than distance ε from L. This calls for another picture, whether mental or real:
Now the definition adds one more qualification: the distance between f(x) and L is less than ε for all values of x that are greater than N. So add this to the picture:
Finally the definition ends by saying that what it has just spelled out is what the “limx→∞f(x) = L” notation means (i.e., that pattern of words and symbols is a shorthand for the previous paragraph of variables and relationships between them).
One last note about reading this definition. It ends by referring you to a figure. Take that advice! Look at the author’s diagram of the situation and see if it agrees with or clarifies yours.
Use the definition of having a limit at infinity to prove that the limit as x approaches infinity of 1/x is 0.
Based on the formal definition, the proof boils down to seeing if, given any ε, I can find a value N such that | 1/x - 0 | < ε whenever x > N. One way to do such “show that I can find...” proofs is to give a process for finding the thing you need. In this case, start by simplifying a little:
Now, we might guess that N could be calculated as 1/ε. As long as this guess produces the result we need, it’s a fine part of the proof. So check that it does produce the right relationships between N, x, ε, etc:
At this point we have shown that it is indeed the case that whenever x > 1/ε, 1/x < ε, which is all we need for the proof.
A useful rule for using derivatives to evaluate certain limits that would otherwise simplify to 0/0 or ∞/∞ — L’Hôpital’s Rule.
Please read “Applying L’Hôpital’s Rule” in section 4.8 of our textbook by tomorrow.
Please also contribute to this discussion of L’Hôpital’s Rule.