Evaluating Limits Analytically (Using Limit Theorems) [Video]


Dave4Math » Calculus 1 » Evaluating Limits Analytically (Using Limit Theorems) [Video]

This video explains the limit theorems, including the Product Rule, Quotient Rule, and the Polynomial Functions Rule, and the Rational Functions Rules. I then work through many examples, including factoring, rationalization, and some limits involving trigonometric functions. In the end, I illustrate the Squeeze Theorem and how to find inequalities when working with the Squeeze Theorem.

In this video, I going to illustrate evaluating limits analytically using the Limits Theorems of Calculus 1. In particular, I will discuss trigonometric limits and finding limits using rationalizations. Then, at the end I will cover the Squeeze Theorem.

Hi everyone, welcome back to my channel, I’m Dave. 

In today’s episode, we concentrate not on the formal definition of a limit of a function but rather give several examples which emphasis algebra and trigonometry techniques to evaluate limits of functions using basic limit theorems. Next I will discuss the squeeze theorem and go through examples showing 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.

Before getting started you may want to review this video here that introduces limits using tables and graphs.

Now let’s get started.

FAQ Evaluating Limits Analytically

What are the 3 methods for evaluating limits?

Using an intuitive approach, for example, tables and graphs is one approach. Another approach is using epsilon-delta style proofs to rigorously find the value of a limit. The third approach, used in this video, is to use algebra and trigonometric limit theorems to evaluate limits analytically.

What is the limit?

Evaluating Limits Analytically Using Limit Theorems

Limits are used to understand the behavior of a function around a point. As the independent variable is approach one value, how are the dependent values behaving? If the dependent values of the functions approach a specific single value, then the limits exists. There are several limit theorems that can be used for evaluating limits analytically.

How do you calculate limits?

Using limit theorems to calculate limits is one way by either using algebraic or other techniques. Or you can use tables and graphs to get an understanding of what the value of a limit might be and then use a proof to validate that guess.

What are the limit rules?

The basic limit rules for calculus 1 are the following: the constant rule, the limit of x rule, the sum and difference rules, and the product and quotient rules. There is also the power rule, the polynomials rule, and the rational rule. The squeeze rule is also very important.


At the beginning of the video I started with the basic limit theorems such as the sum and difference rule for limits. Next, I discuss finding limits using the Quotient rule and the Rational Function rule, these theorems greatly enhance our ability to find limits of these types of functions. I then turned to trigonometric functions where I worked out examples illustrated how to use to very special limits involving trigonometric functions. Then I worked through examples over rationalization and finding limits of piecewise functions. Towards the end I worked through 3 examples demonstrated how to use inequalities and the Squeeze rule to find the value of a limit.

Now I wanna turn it over to you. 

Which do you like more finding a limit using rationalization or using the Squeeze Theorem? Either way let us know what you think in the comments below right now.

If you like this article or the video, please like or leave a comment and subscribe to my channel.

dave at dave4math

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.