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17Calculus Finite Limits - Limits Approaching A Finite Value

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Limits discussed on this page are approaching a constant, i.e. a specific finite number. In limit notation, they look like \(\displaystyle{ \lim_{x \rightarrow c}{~f(x)} }\) where \(c\) is a finite number.

If you want to study the case where \(c\) is infinite, you need to go to the Limits At Infinity (Infinite Limits) page. (The panel below explains the terminology and how 17calculus defines finite and infinite limits.)

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.

Studying limits that approach a constant is the best place to start to get an understanding of how limits work. Make sure you understand this Limit Key before going on.

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 \(c\). That is, \(x\) can get as close as it wants to \(c\) but it will never actually equal \(c\). That seems simple enough but it is extremely important to remember.

However, 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.

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.

Topics You Need To Understand For This Page

basics of limits

General Steps To Evaluate Limits

Step 1 [ substitution ] - - To evaluate a limit at a constant, the first thing you will always want to try is direct substitution. From the Limit Key discussion, you know that a limit doesn't necessarily mean the value AT the point. However, if the function is continuous at the point in question, the limit will be equal to the function at that point. Plugging the number into the function will help you determine if that is the case.
If you plug in the number into the function and you get a finite number without a zero in the denominator, then you are done and that number is your answer.
However, if you get a zero in the denominator, there is more work to do. The next step is determined by what you have in the numerator. If you have a number other than zero in the numerator, then go to Step 2A, otherwise go to Step 2B.

Step 2A ( zero denominator, non-zero numerator ) - - In this case get a non-zero number in the numerator and zero in the denominator. This case means that the limit is either \(+\infty\) or \(-\infty\). In this case you need to look at the behavior of the denominator very near zero to see if it stays negative or positive on both sides of the limit.

Step 2B ( zero denominator AND zero numerator ) - - What's going here is that \(0/0\) is what we call indeterminate. Basically, that means it can't be determined, i.e. \( 0/0 \) can mean anything. It may mean \(0\), it may mean \( 27 \) or \( 19075 \) or infinity. You can't tell. So what we need to do is to work some algebra on the problem to get it in a different form. Usually this means factoring the numerator and denominator to get a common factor that will cancel. [ See the practice problems for demonstrations of this technique. ] Another option may be to use trig identities so that factoring becomes possible.

Step 3 - - Once you have altered the form of the limit, use substitution again to see if anything has changed. If not, then try again by going back to step 2 ( whichever applies ) and try again.

From the above list of steps, we extrapolate four main techniques that you might use, depending on the situation. Here is the list of the techniques.

1. Substitution

2. Factoring

3. Rationalizing

4. Using Trig Identities and Special Trig Limits

Let's start with technique 1, substitution, on the next page.

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