In this article (and accompanying video), I describe limit theorems and illustrate how to use them to find the limit. I also cover trigonometric limits involving sines and cosines and the squeeze theorem. But first, what are limits?
We will begin with what limits are by giving a solid introduction to limits by discussing tables of values and considering graphs of functions.
Then we’re going to talk about limit theorems and also using algebra and trigonometry to find limits. After that, we are going to talk about piecewise functions and the limits of piecewise functions. Finally, after the squeeze theorem, we will work through some examples.
An Intuitive Introduction to Limits
In this article, I begin by discussing one-sided and two-sided limits. I motivate limits intuitively by using tables and graphs. I also explain oscillating behavior and unbounded behavior.
A limit is used to describe the behavior of a function near a point but not at the point. The function need not even be defined at the point. If it is defined there, the value of the function at the point does not affect the limit.
People use limits to study the behavior of quantities under a process of change. For example, we use limits to describe the behavior of a function on its domain. Here we study one-sided limits and two-sided limits with emphasis on graphs. We discuss unbounded behavior and oscillating behavior.
Finding Limits Using Tables
Finding limits using a table of values is an intuitive approach. In this first example, we try to understand how the given rational function is behavior around a given number 3.
The point is, using technology to verify a computation can lead to misunderstanding; and in fact, a formal definition of a limit is needed. Using the formal definition of a limit (to come later in another article), we can prove what the value of the limit is without any doubt.
Finding a limit is impossible, if the limit does not exist
We illustrate when a limit does not exist by giving three case examples.
- A limit does not exist because the one-sided limits do not agree in value.
- A given limit does not exist because of an oscillating behavior of a function, and
- A limit does not exist (no finite value) because of an unbounded behavior of a function.
One-Sided and Two-Sided Limits
Consider, for example, a piecewise function with a jump where the function is pieced together (defined or not). Even though the one-sided limits might exist, they must agree in value for the two-sided limit to exist. In short, if the one-sided limits do not agree then the two-sided limit does not exist.
Limits of Piecewise Functions
Consider, for example, a piecewise function with a jump where the function is pieced together (defined or not). Even though the one-sided limits might exist, they must agree in value for the two-sided limit to exist. In short, if the one-sided limits do not agree, then the two-sided limit does not exist.
Here is an example where a piecewise function is given and we are asked to the find the limit. In some case we are asked to find a one-sided limit and other a two-sided limit.
I then follow this example up with another example just to make sure someone just starting out can better understand one-sided (and two-sided) limits.
In the following example, we find the limit of a piecewise function. Notice that in this example, we do not need to use one-sided limits.
However, in this example, we use onle=-sided limits to find the value of the two-sided limit because the function is behaving differently on each side.
Also, consider another case where a function has an oscillating behavior. On one hand, the trigonometric functions all have an oscillating (periodic) behavior. However, imagine a function where the oscillation becomes much more pronounced as the variable approaches a fixed point; this type of oscillating behavior is where the function may not have a limit.
This last example is worked out in detail in the accompanying video.
Finally, we illustrate the case where a function becomes unbounded as the variable approaches a fixed number; for example, a function with a vertical asymptote. Without a finite number to assign the limit, we say that the limit does not exist.
Finding Limits Using Limit Theorems
Next I demonstrate calculating limits using Limit Theorems. I place importance on examples, especially examples with trigonometric and rational functions. After that, I also discuss rationalization, limits of piecewise defined functions, and the Squeeze Theorem.
As I mentioned at the beginning, in this article, I concentrate not on the formal definition of a limit of a function of one variable but rather give several examples which emphasis algebra and trigonometry techniques to evaluate limits of functions using basic limit theorems.
Here are the basic limit theorems:
For example, these limit theorems say that the limit of a …
- constant is the constant
- sum of functions is the sum of the limits.
- difference of functions is the difference of the limits.
- product of functions is the product of the limits.
- quotient of functions is the quotient of the limits, provided the quotient exists.
- polynomial (rational) function can be found by function evaluation.
Here are some examples demonstrating how to use these limit theorems. In this next example, we apply the quotient rule, the sum and difference rules, and variuous other limit rules.
The note at the end of the next example is important to understand.
In the next example, I illustrate that using the limit theorems should be done with care. In order to use the conclusion of a limit theorems you must first check the hypothesis holds. Here in the next example, the limit of a sum of functions, is not equal to the sum of the limits, because the two-sided limit does not exist.
Compare the previous with the product as follows.
In the next example, we remind ourselves that factoring can be useful, but it is sometimes difficult.
Using Rationalization To Find the Limit
In the next three examples, we use rationalization to help us find the limit. In the first example, we rationalize the numerator, in the second example, we rationalize the denominator. After that, in the third we rationalize with a cubic root.
Special Trigonometric Limits
We use these two limits:
to find the limits of more complex functions that involve other trigonometric functions. Of course, as follows, we have the limit of a trigonometric function as limit theorems also.
Here are six examples demonstrating how to apply these theorems.
This next example is especially fun.
A common technique in trigonometry is to change tangents, cotangents, secants, and cosecants to sines and cosines, then use algebraic simplification and see what happens
Next we state the squeeze theorem and through an example show how to use it. Basically, the idea is to bound a function on both sides by functions whose limits can be more easily computed; and thus in the process squeeze the value of the limit of the original function out.
Let’s consider an example to illustrate this theorem.
In the video I walk you through this example step-by-step, including the behavior of the function around 0. Also in the video I work through these fun exercises (in detail):
Exercises on Finding the Limit
Here are some exercises for you. If you would like me to make a video with the solution to some of these let me know here (in the comments).
In these exercises you will be asked to find the limit.
Calculus 1 (Explore, Discover, Learn) Series
If you would like me to make a video with the solutions to some of the exercises let me know in the comments.
This article (and accompanying video) are a part of a series of articles (and videos) called the Calculus 1 (Explore, Discover, Learn) Series. Also, I put together for you a getting started with calculus 1 page and a video playlist for calculus one.
Have fun in your studies!