# What does it mean when a limit is finite

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• elixer akm
In summary, a finite limit means that there is a real number L to which a series of numbers Rn approaches as n increases, and this difference between L and Rn gets smaller and smaller as n increases. This is true for any given small positive number S, with a corresponding natural number n such that Di < S for all i > n.
elixer akm
What does it mean when a limit is finite

elixer akm said:
What does it mean when a limit is finite
Hi elixer:

I am not sure I understand what kind of answer you are seeking. One possible answer for a series of real numbers Rn is that there is a real number L such that the absolute value difference Dn = |L - Rn| gets smaller and smaller as n increases. One way this is described is that for any given small positive number S, there is some corresponding number n such that Dn < S.

Hope this helps.

Regards,
Buzz

elixer akm
elixer akm said:
What does it mean when a limit is finite

Different types of limits have different definitions.
A "finite limit" generally means one of the following types of limits:

1) ##\lim_{x \rightarrow a} f(x) = L\ ## (e.g. ##lim_{x \rightarrow 3} x^2 = 9##)
2) ##\lim_{x \rightarrow \infty} f(x) = L##
3) ##\lim_{x \rightarrow -\infty} f(x) = L ##

Where "##L##" denotes a number. (The symbols "##\infty##" and "##\infty## do not denote numbers.)

Examples of types of limits that are not finite are:
4) ##\lim_{x \rightarrow a} f(x) = \infty\ ## (e.g ##lim_{x \rightarrow 0} \frac{1}{x^2} = \infty##)
5) ##\lim_{x \rightarrow a} f(x) = -\infty##.
6) ##\lim_{x \rightarrow \infty} f(x) = \infty##
7) ##\lim_{x \rightarrow \infty} f(x) = -\infty##
8) ##lim_{x \rightarrow -\infty} f(x) = \infty##
9) ##lim_{x \rightarrow -\infty} f(x) = -\infty##.

Since many different types of limits are defined, the terminology can get confusing. Some textbooks would say cases 4),5),6),7),8),9) are cases where "a limit does not exist". Nevertheless, those types of limits are useful in discussing problems, so most textbooks use some terminology for them. Some books call those cases "improper limits".

elixer akm
elixer akm said:
What does it mean when a limit is finite

Hi, at first this tell you that the limit exists, second that is not infinity (or in other terms ''uncontrolled''). In this case when you are in a very small neighborhood of your point the function remains bounded in a finite region and tends to a value ##L## (positive, negative or ##0##), this is true every time you are much near to the point so that we say that ##f## tends to a value ##L## when ##x## tends to a finite point ##x_{0}##. The rigorous definition of a finite limit can be founded in a math book of high school and I hope to have clarified the general idea ...
Ssnow

Buzz Bloom said:
Hi elixer:

I am not sure I understand what kind of answer you are seeking. One possible answer for a series of real numbers Rn is that there is a real number L such that the absolute value difference Dn = |L - Rn| gets smaller and smaller as n increases. One way this is described is that for any given small positive number S, there is some corresponding number n such that Dn < S.
Should say "natural number n such that Di < S for all i > n". So the sequence of numbers gets arbitrarily close to the limit number and stays close from then on.

Buzz Bloom

## 1. What is a finite limit?

A finite limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. It means that the value of the function at that point is well-defined and does not approach infinity or negative infinity.

## 2. How do you determine if a limit is finite?

To determine if a limit is finite, you can evaluate the function at the given input value and see if the result is a real number. If the result is a real number, then the limit is finite.

## 3. Can a limit be both infinite and finite?

No, a limit cannot be both infinite and finite. It can either be one or the other. If a limit is infinite, it means that the function has a vertical asymptote at that point, and if it is finite, it means that the function is continuous at that point.

## 4. What does it mean when a limit is approaching zero?

When a limit is approaching zero, it means that the input value is getting closer and closer to zero. This does not necessarily mean that the limit is finite, as the function could still approach infinity or negative infinity at that point.

## 5. What is the importance of a finite limit in mathematics?

A finite limit is important in mathematics because it helps us understand the behavior of a function at a specific point. It allows us to determine if the function is continuous or has a vertical asymptote at that point, which can help us make predictions and solve problems in various fields such as physics, engineering, and economics.

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