PreCalculus - a basic look at limits

Another one of the properties of functions that we cover is describing end behavior. And we have students write this in limit notation. Without really going over what a limit is technically. Just teaching the notation.

This year I decided to do a little mini-lesson on limits to prepare for deeper understanding of the limit notation for students.

We've already done "limits" informally without using that language when we looked at asymptotes and especially when we examined "holes" in rational functions more closely. I always have the students figure out the point of discontinuity for those "holes" by choosing values close to the offending x value. We do this on our graphing calculators on the graph using the "trace" option of finding function values with x-values that we type in. If the hole is at x = 1 students will only see [x = 1 y =   ] on the screen. So we try values "around" one. We try x = 0.9999 and x = 0.0001. All very limit like without saying so.

I set the stage by reminding them of what asymptotes are and what we did when we found those holes. Also we reviewed what students learned about end behavior in Algebra 2. They did learn this concept but had another way to describe it with LEB and REB. Not limit notation, but has some similarities.

Then we go into defining and exploring with a limit is. I use this "french-door" foldable to do this.

We do "What is a limit" and then "what is limit notation". Then we do the geometric visual (love this) and then the classic limit example.

Then we do the two examples. The first one does have a value at the given value of x, so I tell them we can use "substitution" to find that limit.

The second one can not be evaluated at the given value of x, so we examine the value using the table function of our graphing calculator.


We first change our table set up so the independent variable is Ask. [2nd][window]

Then we type the equation in Y1 [y=]

Then we go to the table [2nd][graph]. Clear any values you might see with [del] (delete).

Now with a fresh empty table we can type values in close to our given x. I tell them to go ahead and type in x = 1, they get a rude little ERROR in the table.

The students see from the table that values get closer and closer to 2 as the values of x get closer and closer to 1. So that's our limit. Yay!

Then they get a little homework of finding other limits the same way - either substitution or with the table. Next class - End Behavior.

Foldable documents here.