# Formal Def of Limit: Why is |x-c| < δ and not |x-c| <= δ

In summary: D - \{p\}##" means the "set D with the point ##p## removed". ) .I hope this gives you some idea of the reason why the epsilon-delta definition is stated as it is.In summary, the conversation discusses the mathematical representation of the limit of a function at a specific point. The correct definition uses "less than" instead of "less than or equal to" to exclude the case where the input value is equal to the specific point. This is a convention in advanced mathematics and is based on the use of open sets and mappings between sets. The use of "less than" does not make a difference in the definition and it is a matter of culture and tradition.
Homework Statement
Taking notes on the formal definition of limits.
Relevant Equations
lim (as x -> c ) = L
δ > 0 ; ϵ > 0
*Given: δ = |all real numbers|
*Given: ϵ = |all real numbers|

For any x value within +/- |δ| of c, we can find a y=f(x) within the corresponding +/-|ϵ| of L. According to my professor, the mathematical representation of this is |x-c| < δ and |f(x) - L | < ϵI fail to understand why it cannot be equal to ϵ or δ.

Is it because this all applies when x=/=c but gets infinitely closer to it.

There is no mathematical reason, they both work. The reason why "<" is the common case, is that we deal with open sets here, so the definition goes without the boundary. The most common definition of continuity in topological spaces (which are defined by stating what the open sets are) is that the preimage of any open set is open. And open in ##\mathbb{R}^n## means "<". It is a convention rather than a necessity.

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Is it because this all applies when x=/=c

Yes.

The following statements are all equivalent:

$$\forall \epsilon> 0: \exists \delta > 0: \forall x: 0 < |x-c| < \delta \implies |f(x)-L| < \epsilon$$
$$\forall \epsilon> 0: \exists \delta > 0: \forall x: 0 < |x-c| \leq \delta \implies |f(x)-L| < \epsilon$$
$$\forall \epsilon> 0: \exists \delta > 0: \forall x: 0 < |x-c| < \delta \implies |f(x)-L| \leq \epsilon$$
$$\forall \epsilon> 0: \exists \delta > 0: \forall x: 0 < |x-c| \leq \delta \implies |f(x)-L| \leq \epsilon$$

This is actually a very good exercise to see that you understand how quantors work, so you should try to prove hese equivalences.

Notice if the result applies when ## |x-c|< \epsilon## it must also apply in any closed subinterval by definition ,e.g., in ##|x-c| \leq \epsilon/2##. It also ties down with the definition of continuity at a point ##x## :for every open 'hood ##U_{f(x)}## of ##f(x)## there is a 'hood ## V_x## of ##x## with ##f(V_x)\subset U_{f(x)}##

According to my professor, the mathematical representation of this is |x-c| < δ and |f(x) - L | < ϵ

The correct definition uses ## 0 < |x -c| < \delta## This condition excludes the case ##x = c##.

For example, define the function ##f(x)## by:
##f(x) = 2x ## if ##x \ne 3##
##f(3)= 47##

The way the definition ##\lim_{x \rightarrow 3} f(x)## is written, we have ##\lim_{x \rightarrow 3 } f(x) = 6 ##.
If we wrote the definition using "##|x - c| < \delta##" , a person could argue as follows.

##lim_{x \rightarrow 3} f(x) ## is not ## L = 6##, because for ##\epsilon = 0.1## , the value ##x = 3## does not satisfy ##| f(x) - 6| < 0.1##. We have ## |3 - 3| = 0 < \delta## , but ##|f(3) - 6 | = |47 - 6| = 41 > 0.1 ## no matter how small ##\delta > 0 ## is chosen.

If this is the first time you have encountered limits of functions, you might prefer that ##\lim_{x \rightarrow 3} f(x) ## not be 6 because ##f(x)## is not 6 when ##x## is exactly 3. However, it has proven useful in mathematics to define the ##lim_{x \rightarrow c} f(x) ## in such a way that it does not depend on a value at ##f(c)##.

For example, using the correct definition, we can consider limits such as ##\lim_{x \rightarrow 0} \frac{\sin(x)}{x}## where the function has no defined value at ##x = 0##.
For any x value within +/- |δ| of c, we can find a y=f(x) within the corresponding +/-|ϵ| of L.
The way you expressed the idea ("within"), you didn't exclude the case ##x = c##.

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I fail to understand why it cannot be equal to ϵ or δ.

As other replies indicate, using "##\leq##" instead of "##\lt##" wouldn't make any difference.

The reason "##\lt##" is used is matter of culture and tradition. In advanced mathematics, we consider functions that are "mappings" between sets. Introductory Calculus deals with functions such as ##f(x)= x^2## , which are mappings from a real number to another real number. There can be mappings of a more general type. For example the function defined by ## F((x,y)) = (2x,xy)## maps a point ## (x,y)## in 2 dimensional space to another point in two dimensions. In the more general case, we define limits in terms of "open sets" of points. If you are familiar with the notation "(a,b)" to indicate an "open interval" on the real number line, you have some hint of the use of "open sets".

In the general case of mappings, the definition of ##lim_{x \rightarrow p} F(x) = L## can be expressed as:
For each open set ##E## containing the point ##L## , there exists an open set ##D## containing the point ##p## such that ##F(D - \{p\}) \subset E##. (The notation ##F(D - \{p\})## represents the set ##\{F(x): x \in D, x \notin\{p\}\}## ).

So, in introductory calculus, it is traditional to phrase the definition of ##\lim_{x \rightarrow c} f(x)## so it can be regarded as a statement involving "open sets". This makes the definition follow the same pattern as the definition for limit of a mapping used in advanced mathematics. To use "open intervals", we must use "##\lt##" instead of "##\leq##".

For example, for ##\delta > 0##, ##\{x: |x-c| \lt \delta\}## represents the open interval ##(c - \delta, c + \delta)##. By contrast, ##\{x: |x-c| \leq \delta\}## represents the "closed interval" ##[c - \delta, c + \delta]##.

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## 1. What is the formal definition of a limit?

The formal definition of a limit is a mathematical concept used to describe the behavior of a function as the input approaches a certain value. It states that the limit of a function f(x) as x approaches a value c is equal to L if for any positive number ε, there exists a positive number δ such that whenever the distance between x and c is less than δ, the distance between f(x) and L is less than ε.

## 2. Why is the absolute value of the difference between x and c less than δ in the formal definition of a limit?

The absolute value is used to ensure that the distance between x and c is always positive, regardless of whether x is larger or smaller than c. This is important because the value of δ should be positive and cannot be negative.

## 3. What does the inequality symbol < symbol in the formal definition of a limit represent?

The symbol < represents "less than" in the formal definition of a limit. It is used to show that the distance between x and c must be strictly less than δ in order for the limit to hold.

## 4. Why is the strict inequality < used instead of the inclusive inequality <= in the formal definition of a limit?

The strict inequality < is used because it ensures that the limit only holds for values of x that are strictly closer to c than δ. If the inclusive inequality <= was used, it would allow for values of x that are equal to c, which may not necessarily hold true for the limit.

## 5. What is the significance of the value of δ in the formal definition of a limit?

The value of δ represents the maximum distance between x and c for which the limit is guaranteed to hold. It is an important factor in determining the behavior of a function as x approaches c, and it can vary for different functions and values of c.

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