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WikiBooks - Calculus/Formal Definition Of The Limit

Here is a page containing examples (and solutions) showing how to use the Limit Definition to prove a limit. UC Davis - Limit Definition Examples.

Limits FAQs

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Limits form the basis of all of calculus. So it is important to understand and be able to use limits. This topic is going to stretch your mind a bit but if you stick with it, you will get it.

Finite Limits, Infinite Limits, Limits At Infinity . . . Terminology Explained

The use of the terms finite limits, infinite limits and limits at infinity are used differently in various books and your instructor may have their own idea of what they mean. In this panel, we will try to break down the cases and explain the various ways these terms can be used as well as how we use them here at 17calculus.

When we talk about limits, we are looking at the \(\displaystyle{ \lim_{x \to c}{f(x)} = L }\). The various terms apply to the description of \(c\) and \(L\) and are shown in the table below. The confusion lies with the terms finite limits and infinite limits. They can mean two different things.

\(\displaystyle{ \lim_{x \to c}{f(x)} = L }\)

when

term(s) used

\(c\) is finite

limits approaching a finite value or finite limits

\(c\) is infinite \(\pm \infty\)

limits at infinity or infinite limits

\(L\) is finite

finite limits

\(L\) is infinite \(\pm \infty\)

infinite limits

You can see where the confusion lies. The terms finite limits and infinite limits are used to mean two different things, referring to either \(c\) or \(L\). It is possible to have \(c = \infty\) and \(L\) be finite. So is this an infinite limit or a finite limit? It depends if you are talking about \(c\) or \(L\).

How 17calculus Uses These Terms
The pages on this site are constructed based on what \(c\) is, i.e. we use the terms finite limits and infinite limits based on the value of \(c\) only ( using the first two rows of the table above and ignoring the last two ). This seems to be the best way since, when we are given a problem, we can't tell what \(L\) is until we finish the problem, and therefore we are unable to determine what type of problem we have and know what techniques to use until we are done with the problem.

Important: Make sure and check with your instructor to see how they use these terms.

Limit Notation

First, let's talk a bit about limit notation. To understand why we need to discuss notation, read this next panel on why notation is important in calculus.

Why Notation is Important in Calculus

Using correct notation is extremely important in calculus. If you truly understand calculus, you will use correct notation. Take a few extra minutes to notice and understand notation whenever you run across a new concept. Start using correct notation from the very first.

You may not think this is important. However, if your current (or previous) teacher doesn't require correct notation, learn it on your own. You may (and probably will) get a teacher in the future that WILL require correct notation and this will cause you problems if you don't learn it now. It is much easier to learn it correctly the first time than to have to correct your notation later, after you have been doing it incorrectly for a while.

Not only is it important in class to use correct notation, when you use math ( or any other subject that uses special symbols ) in your career, you will need to be able to communicate what you mean. Without correct notation, your ideas could be misunderstood. It's a lot like speaking English ( or whatever language you use regularly ) or speaking a variant that has meaning only to you. You will be misunderstood and it may even affect your ability to keep a job.

So, just decide to start using correct notation now. It's not that hard and it will pay off in the long run.

When writing a limit, we use the notation \(\displaystyle{ \lim_{x \rightarrow c}{~f(x)} }\). There are three important parts to this notation that must all exist for this notation to have meaning.
1. The three letter abbreviation 'lim'. This tells you that you have a limit.

2. The notation \( x \rightarrow c \). This is important because it tells you two things; the variable that the limit applies to, 'x', and the value it is approaching, 'c'. Leaving this off when using limit notation leaves the reader guessing the variable and the value.

3. The function itself, 'f(x)'. Without this you don't know what function the limit is being applied to.

It is important that you have all three parts at all times when you use this notation. A good calculus teacher will encourage this by taking off points if you leave off any part.

Note: In type-written documents, including your textbook, you may see the limit written as \( \lim_{x \rightarrow c}{~f(x)} \). It is written this way to save space. However, it is not okay to write it this by hand. You need to write it as \(\displaystyle{ \lim_{x \rightarrow c}{~f(x)} }\) with the \(x \to c\) directly underneath lim.

Basic Idea of Limits

Now that you understand the notation related to limits, let's watch a few videos explaining the basic idea of limits and how to work with them. It is important to watch the first two videos to get a full understanding.

This first video is a great introduction video explaining the basic idea of limits and how to get started.

PatrickJMT - What is a Limit? Basic Idea of Limits [5mins-6secs]

video by PatrickJMT

This next video clip has a great explanation of limits using some examples.

Krista King Math - Limits and Continuity [8mins-41secs]

Notes:
1. The second half of this video discusses continuity, especially discontinuities. It is best to watch it in context on the continuity page.
2. In general, this is a good video. However, she says something incorrect near the end of the video. When she is discussing a limit at infinity and she is looking at three cases of rational functions, she says, in the third case, that if the degree of the numerator is greater than the degree of the denominator, then the limit will be positive infinity when the limit goes to positive infinity and the limit is negative when the limit goes to negative infinity. That is not always the case and it is much more complicated than she implies. You need to look at the sign of the coefficients and the relationship between the highest powers. There are lots of practice problems on the infinite limits page that show what can happen. Other than this, the rest of the video is very good.

video by Krista King Math

The rest of this page contains a few topics related to limits in general. To study specific topics in depth, you can find links in the menu above. These individual pages contain discussion, videos and practice problems. However, before you go there, the next few sections contain important background information you need to understand the other topics. So it will help you to go through these sections.

Limit Key

One key that you need to remember about limits is when you use the limit notation \[ \lim_{x \rightarrow c}{ ~f(x) } \] this means that x APPROACHES c but is never equal to c. This means x can get as close as it wants to but it will never actually equal c. That seems simple enough but it is extremely important to remember. There are many times when you can determine the limit by substituting x for c to calculate f(c). However, those are special cases that require special conditions and is not true in every case. There is probably a theorem in your textbook that tells the special conditions that must exist for you to be able to apply this.

Here is an explanation of how this concept is written mathematically. Look more carefully at the definition of the limit at the top of the page. Notice that it requires \( \delta \gt 0 \). This means that \( |x-c| > 0 \) and, therefore, x can never equal c.

I know this seems like a minor point, but it isn't. If you remember this, you will have a good start on your way to understanding limits.

Basic Limit Laws

In general, limits follow the algebra rules as you would expect. However, as with all of higher math, we need to explicitly write out the rules, so that we know what we can do and what we can't do. So, here are some basic limit laws.
Note: In all cases below, when we state that a variable is a constant, that means it is a real number and not \(\infty\) or \(-\infty\). However, these rules also apply when \( x \to \infty \) or \( x \to -\infty \).
In these equations, we assume that
1. \(b\), \(c\) and \(k\) are constants
2. \(\displaystyle{ \lim_{x\to c}{~f(x)} = L }\) and \(\displaystyle{ \lim_{x\to c}{~g(x)} = M }\)
3. \(n\) is a positive integer

Constant Rule

\(\displaystyle{ \lim_{x\to c}{ ~b } = b }\)

Identity Rule

\(\displaystyle{ \lim_{x\to c}{~x} = c }\)

Power Rule

\(\displaystyle{ \lim_{x\to c}{[f(x)]^n} = }\) \(\displaystyle{ \left[ \lim_{x\to c}{~f(x)} \right]^n }\)

Operational Identities

1. Constant Multiple Rule

\(\displaystyle{ \lim_{x \to c}{[k\cdot f(x)]} = }\) \(\displaystyle{ k \cdot \lim_{x \to c}{f(x)} = kL }\)

2. Sum/Difference Rule

\(\displaystyle{ \lim_{x\to c}{[f(x) \pm g(x)]} = }\) \(\displaystyle{ \lim_{x\to c}{~f(x)} \pm \lim_{x\to c}{~g(x)} = }\) \(\displaystyle{ L \pm M }\)

3. Multiplication Rule

\(\displaystyle{ \lim_{x\to c}{[f(x) \cdot g(x)]} = }\) \(\displaystyle{ \lim_{x\to c}{~f(x)} \cdot \lim_{x\to c}{~g(x)} = }\) \(\displaystyle{ L \cdot M }\)

4. Division Rule

\(\displaystyle{ \lim_{x\to c}{[f(x)/g(x)]} = }\) \(\displaystyle{ \frac{\lim_{x\to c}{~f(x)}}{\lim_{x\to c}{~g(x)}} = }\) \(\displaystyle{ \frac{L}{M}, ~~~ M \neq 0 }\)

Here is a video discussing these properties and giving examples to help you understand how to use them.

MIP4U - Properties of Limits [7mins-29secs]

video by MIP4U

Limits of Composite Functions

When you have a composite function, the limit works as you would expect as long as some conditions are met.

Theorem: Limit of a Composite Function

If we have functions \(f(x)\) and \(g(x)\) and \(\displaystyle{ \lim_{x\to c}{g(x)} = L }\) and \(\displaystyle{ \lim_{x\to c}{f(x)} = f(L) }\) then
\(\displaystyle{ \lim_{x\to c}{f(g(x))} = f(L) }\)

Note in the above theorem, we can write \(\displaystyle{ \lim_{x\to c}{f(g(x))} = f\left( \lim_{x\to c}{g(x)} \right) }\), i.e. we can move the limit inside the function \(f\).
Not explicitly stated in the theorem are the following assumptions:
1. \(\displaystyle{ \lim_{x\to c}{g(x)} }\) exists - if the limit did not exist, then we could not write \(\displaystyle{ \lim_{x\to c}{g(x)} = L }\)
2. \(L\) is in the domain of \(f(x)\) and is finite - if \(L\) is not in the domain of \(f(x)\) or \(L\) is infinite then \(f(L)\) has no meaning
Okay,let's watch a video proving this theorem.

Larson Calculus - Proof - The Limit of a Composite Function [2min-9secs]

video by Larson Calculus

This next video contains a short explanation and some examples to help you understand this concept.

Krista King Math - Limits of composite functions - How to find them? [22min-5secs]

video by Krista King Math

Difference Between A Limit Going To ±Infinity and A Limit That Doesn't Exist

Many people get confused between the case when a limit goes to infinity (or negative infinity) and when a limit does not exist. And there is a good reason for the confusion. Many discussions I've read do not separate the two cases. But they are different.

when a limit DOES exist

When a limit goes to positive or negative infinity, the limit DOES exist. The limit is exactly that, positive or negative infinity.

when a limit DOES NOT exist

There is only one case when a limit doesn't exist: when the limit is different from the left than it is from the right. This concept requires understanding one-sided limits. There are videos on that page showing examples of when the limit doesn't exist.

For more detail including graphs, see the substitution section on the finite limits page. And, as usual, check with your instructor to see how they define limits that do not exist.

faq: What does 'indeterminate' mean?

The word 'indeterminate' usually shows up when discussing limits and it means 'cannot be determined'. For example, when you take a limit and get the result \( 0/0 \), this is indeterminate, meaning the limit could be anything and cannot be determined with the given information.

Many times, you can determine the limit but in order to do so, you need to alter the function using algebra, trig or L'Hôpital's Rule to put it in a form so that is no longer indeterminate. You can find plenty of examples (practice problems) on the finite limits page, infinite limits page and L'Hôpital's Rule page.

You can find more information on 17Calculus: Indeterminate Forms (including a table listing most indeterminate forms you will see in calculus).

Okay, now you are ready to go to the next topic:
the epsilon-delta (formal) definition of the limit.

next: ε-δ limit definition →
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