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Limits: Vital Building Blocks of Calculus

by blackspanielgallery

Finding limits in mathematics is important in understanding calculus. This article has several videos that will help make this calculus topic understandable.

A limit is simple to understand, it is the value a function gets arbitrarily close to at a specified value of the independent variable. It is not necessary that the function be defined at that value, so I will not call it a domain value of the function.

Translating to English from math, if a graph of a function through a point would exist, then the function’s value, f(x), that makes the graph continuous through that point is its limit as the x value of the point is approached. We literally ignore what the function does at the x-value that is being approached.

For a continuous function the limit at a value is simply found by evaluating the function at that value. For continuous functions it is acceptable to discuss the limit at a domain value.

The limit for a function with a hole in the domain, such as a hole caused by a division by zero, can still exist if the function would be continuous if the hole in function’s graph were to be filled in. This also works even if there is a functional value that does not fill in the hole. Simply use the value as the limit that would make the function continuous.

The Details Are in the Videos

This article is to bring several videos, each focused of a different aspect of limits and their use, into a flow of the concept. 

 

I am adding more math tutorial videos, so subscribing to my channel will allow you to find these and other videos easily.  There are already other calculus videos available.

Finding Limits

What Happens if the Function Tends to Infinity, Or Negative Infinity?

It is not possible to fill in a hole in a domain using infinity, but if a function tends to increase, or decrease, without bound from the left and from the right of a missing domain value, the limit is said to be infinity or negative infinity depending on which is being approached. 

How Do Limits Impact Calculus?

Calculus has an important concept, the derivative.  The derivative allows finding the slope of the tangent line of the function at a point.  The theory is that the limit of secant line segments, or chords, approaches the slope of the tangent line as the two points defining the slope of the secant lines become very close.  This results from the points (x + h, f(x + h) and (x, f(x)) becoming very close as h approaches zero. 

 

A second important concept of calculus is the integral, or the antiderivative.  The integral gives the “area” under a curve, with the understanding that in this sense “area” can be negative.  This is possible because we can partition the area into very narrow rectangles, with the width of the rectangles having a limit of zero.  The rectangles then have their areas added.  The use of a limit is very important here.

 

Limits form building blocks for both derivatives and integrals, two major concepts of calculus.  While it is possible to avoid using limits in differentiating a function or in integrating a function, limits are at the core of the theory that makes our shortcut formulas possible.

Derivatives

What Happens at Infinity, Or at Negative Infinity?

Just as a function can increase or decrease without bound, so can the domain.  What happens as x approaches infinity, or approaches negative infinity?  We cannot approach infinity from both sides, but we can approach negative infinity from the right and infinity from the left.  By taking one-sided limits we can see what happens at the extremes of the function’s domain.

Limits at Infinity and at Negative Infinity

L'Hopital’s Rule

Also Spelled L'Hospital's Rule

For limits that result in zero over zero, or an undefined expression over an undefined expression, we can often use L’Hopital’s rule to find the limit.  This involves taking the derivatives of the numerator and of the denominator, and seeing if that resolves the difficulty.  It is possible it will not, but it is acceptable to apply L’Hopital’s rule multiple times if necessary.

L'Hopital's Rule

Limits Are an Important Tool in Understanding

Limits are not just a way to avoid problems such as divisions by zero, they are important to the theory in two major concepts of calculus. 

 

Calculus is so much easier if we understand why it works, instead of memorizing many formulas with no idea what makes them valid. 

More to Come

I am planning to add other articles with more videos embedded.  These will be forthcoming. 

 

This article contains links to affiliate programs and Adsense advertising.  These must use cookies to allow for proper crediting. As an Amazon Associate I earn from qualifying purchases.

Updated: 08/02/2024, blackspanielgallery
 
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DerdriuMarriner on 08/17/2024

Thank you for your comment below in answer to my previous observation and question.

The two kinds of calculus books both interest me, what with my social-science background and what with my business and life-science experiences.

The free-books option is most interesting too and therefore much appreciated!

blackspanielgallery on 08/17/2024

If you are looking for a calculus book there are two kinds. One is calculus, the other is calculus for business and life science majors. The second is in many cases light on theory and does not reach such topics as line and surface integrals, or even finding volumes. It covers a single three hour course, and does not even have trig functions. It does a good job on covering the basics. As for cost, free is possible using an online textbook. I am familiar with OpenStax, a division of Rice University, that is a collection of free textbooks. I believe a grant might have been used to pay for the writing, but that is a guess. Colleges are going to free books. A nominal fee is charged to get the ability to print one, perhaps as low as a few dollars, and you use your paper and ink. They have free books in many areas of study, check them out.

DerdriuMarriner on 08/17/2024

Thank you for your comment below in answer to my previous observation and question.

Some self-instructees, such as myself, appreciate the non-digital, old-fashioned self-learning through appropriate books.

Is there such a thing as a one-stop calculus book (to serve not as replacement or supplement but as reinforcement and review along with your video series)?

blackspanielgallery on 08/07/2024

All. I do not limit the potential audience.

DerdriuMarriner on 08/07/2024

Thank you for your comment below in answer to my previous observation and question.

Those learning, reviewing or stumbling over geometry undoubtedly count among those going to go through your math videos as they get released!

Is the geometry video going to be for distance-learners, home-schoolers and self-instructees?

blackspanielgallery on 08/06/2024

Yes, I probably will have some videos on geometry as well. There certainly must be a discussion on similar triangles before getting into trig.

DerdriuMarriner on 08/06/2024

Thank you for your comment below in answer to my previous observation and question.

Algebra, trigonometry and probability sound most welcome.

Would there be a demand or a need for geometry?

blackspanielgallery on 08/05/2024

I plan to add algebra, many videos are already done, trig which is also started, probability, also started. and plan to cover most freshmen and sophomore college math, as well as physics. Since the Calculus and physics are the most popular, I am working there right now, but do occasionally handle random college math courses. The articles are to come out when enough related material on related topics make writing a long enough article to publish. I am getting ready to add derivatives.

DerdriuMarriner on 08/05/2024

Thank you for your comment below in answer to my previous observation and question.

Making videos is so proactive (Best of good luck ;-D), timeless and timely.

The calculus videos must be quite a boon for those who muster distance-learning with many other students or self-instructing.

What would be the other math branches that would be upcomingly released?

blackspanielgallery on 08/03/2024

I intend them to assist any student that needs the help. I get to the point and try to make them understandable. Often math is not clearly explained by some professors, so there is a need.

Making videos is my new endeavor. I have others, but I am going to collect groups of related videos and embed them in articles. I also have been revising old articles to include my videos, including other branches of math, physics videos, and even others on Mardi Gras.

My hope is to get channel subscribers, which is needed before I can earn from my videos.

I appreciate the comment.

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