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.
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.
\(\displaystyle{ \lim_{x \to c}{f(x)} = L }\) | ||
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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.
If you want a complete lecture on this topic, we recommend this video.
video by Prof Leonard |
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Recommended Books on Amazon (affiliate links) | ||
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Getting Started With Limits
Before we get into the details of working with limits, let's watch these videos to get an overview of where we are going. There is a lot of detail here but these are worth watching.
video by Dr Chris Tisdell |
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video by Dr Chris Tisdell |
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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.
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.
video by PatrickJMT |
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This next video clip has a great explanation of limits using some examples. First, some comments are in order.
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 |
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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.
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.
video by MIP4U |
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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 |
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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 |
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.
video by Larson Calculus |
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This next video contains a short explanation and some examples to help you understand this concept.
video by Krista King Math |
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Limits of Continuous Functions
You may run across limits of functions that are not polynomials, such as natural logs and trig fuctions. As long as the function is continous on it's domain, you can move the limit inside the function. For example,
\(\displaystyle{ \lim_{x \to 1}{ \ln(x) } = \ln \left[ \lim_{x \to 1}{x} \right] = \ln(1) = 0 }\).
Since the natural log function is continuous on it's domain, we can move the limit inside the natural log function and substitute in for \(x\).
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.
Okay, now you are ready to go to the next topic, the epsilon-delta (formal) definition of the limit.
You CAN Ace Calculus
external links you may find helpful |
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Here is a page containing examples (and solutions) showing how to use the Limit Definition to prove a limit. UC Davis - Limit Definition Examples. |
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
---|---|---|
\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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