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17calculus > limits |
Topics You Need To Understand For This Page
Calculus Main Topics
Limits |
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Single Variable Calculus |
Multi-Variable Calculus |
Tools
math tools |
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general learning tools |
additional tools |
Related Topics and Links
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. |
Limits FAQs
Limits | ||||||||||||||||||
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on this page: ► limit notation ► basic idea of limits ► limit key ► limit laws ► difference between a limit going to ±∞ and a limit that doesn't exist | ||||||||||||||||||
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.
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 \(L\). |
Limit Notation |
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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. |
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 |
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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 | |
This next video clip has a great explanation of limits using some examples.
Krista King Math - Limits and Continuity | |
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 |
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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.
Limit Laws |
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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 |
\(\displaystyle{ \lim_{x\to c}{~x} = c }\) |
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 |
\(\displaystyle{ \lim_{x\to c}{[f(x)/g(x)]} = }\) \(\displaystyle{ \frac{\lim_{x\to c}{~f(x)}}{\lim_{x\to c}{~g(x)}} = }\) \(\displaystyle{ \frac{L}{M} ~~~ M \neq 0 }\) |
Power Rule |
\(\displaystyle{ \lim_{x\to c}{[f(x)]^n} = }\) \(\displaystyle{ \left[ \lim_{x\to c}{~f(x)} \right]^n }\) |
Difference Between A Limit Going To ±∞ and A Limit That Doesn't Exist |
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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, which you can find on this page. There are also 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 definition of the limit. |
next: \(\epsilon\)-\(\delta\) limit definition → |
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faq: What does 'indeterminate' mean?
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).