What does it mean when a limit is finite

[elixer akm]

What does it mean when a limit is finite

 

[Buzz Bloom]

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 R_n is that there is a real number L such that the absolute value difference D_n = |L - R_n| 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 D_n < S.

Hope this helps.

Regards,
Buzz

 

[Stephen Tashi]

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".

 

[Ssnow]

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

 

[FactChecker]

Hi elixer:

I am not sure I understand what kind of answer you are seeking. One possible answer for a series of real numbers R_n is that there is a real number L such that the absolute value difference D_n = |L - R_n| 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 D_n < S.

Should say "natural number n such that D_i < S for all i > n". So the sequence of numbers gets arbitrarily close to the limit number and stays close from then on.