In this chapter, we will introduce the first big idea in AP Calculus: Limits and Continuity. This a topic that is included in both the AB and BC Calculus courses.
Even if your course hasn’t started yet, a good way to prepare yourself for it is to study limits, as they are generally easy to grasp. This will give you a head start on (most likely) the first topic you will learn in your AP calculus course, and can inhibit you from falling behind.
Let’s get started!
A limit is written in the form
where f(x) is a function. This is read “the limit as x approaches c of f of x.”
Here, c is a constant, but note that it can be infinity or negative infinity as well and would be written as
Let’s look at a visual example. In the graph below, the blue graph represents a function. Take a look at the point x = 1 marked in green.
Clearly as the graph approaches x = 1 from both the left and right sides of the point, the function gets closer and closer to the value f(x) = 0.
Since the limit from both sides of the point approach the same value of f, we can write
There is also a special way to write the limits from the right and left. These are called one-sided limits and are written as:
Notice that the limit as x approaches f(x) from the right uses a superscript “+” sign next to the c and the limit as x approaches f(x) from the left uses a superscript “-“ sign next to the c.
Knowing that the one-sided limits at a point are defined and equal, we can conclude that the limit as x approaches that value is defined and equal to that same value. Back to the example above, we can write:
This one-sided limit notation may seem useless at first but that’s because we’ve only looked at one example that was… well, simple.
Here’s a harder example.
Hmm… this graph looks different from the other one we saw. The first thing we notice is that there are gaps or breaks at x = -2 and x = 2. These are called jump discontinuities. Jump discontinuities occur when both one-sided limits at a point exist but are different.
At x = 2, we have
And since the one-sided limits do not match up, our limit as x approaches 2 does not exist.
Notice that even though the function f(x) never took on the value of 2.5 (as indicated by the open circle at the point), we were still able to express the one-sided limits at the point, as limits denote the value that a function approaches rather than the actual value it takes on.
Similarly, the limit as x approaches x = 2 (another jump discontinuity) does not exist.
At x = 4, something weird happens to our function. The value of f(x) seems to get larger and larger without bound as x approaches 4 from the left and seems to get really negative without bound as x approaches 4 from the right. We call this an infinite discontinuity, which occurs when both one-sided limits approach infinity or negative infinity.
We notate this with:
Next, let’s see how we would identify the limits at points without using a graph.
Let g(x) = 2x + 1. Let’s look at the values of g(x) as x gets very close to 3 from the left. In other words, we are looking for the limit as x approaches 3 of g(x) = 2x + 1 from the left.
|x||g(x) = 2x + 1|
I think you can figure out what would happen if we continue the table for values of x that are super close to 3. Right! The values of g(x) are approaching 7, so we would expect the values of g(x) to get extremely close to 7.
Similarly, if you looked at a table using values of x that approached 3 from the right, you would see that g(x) gets extremely close to 7.
Thus, we have:
You can check that this is true by graphing g(x) = 2x + 1 or by simply plugging in the value of 3 for x. Either way we get that g(3) = 7 which corroborates our finding of the limit.
However, it is not always true that the limit as x approaches some value c of a function equals the evaluated value of c of that function. In the example above, we did see equality, but this is not always the case. We can only hope to estimate the value of the function at c using the limit, and generally we get a pretty good estimate.
If it is not clear to you why this is, it will make more sense when we talk about continuity.
In our example above, the limit was equal to the evaluated value at x = 3, because our function was continuous at x = 3.
Limits are very useful tools for evaluating functions, and they are used to a great extent in calculus. Though they seemed pretty easy in this lesson, they can get more challenging and are not always so straight-forward. Sometimes, manipulations are required to find the limit at a point.
Make sure you completely understand the general concept of limits before moving on to more difficult topics as they build upon this big idea.
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Excellent lesson, Hannah! 🙂