The real number Delta

**novice** · July 17th 2010, 08:16 PM

If $f:X \rightarrow \mathbb{R}$ is defined by $f(x)=\frac{x}{x^2-1}$ , and we are interested in the behavior of $f$ near $0$ , then $1,-1 \not\in X$ . A natural choice of $X$ is $\mathbb{R}-\{-1,1\}$ , in which case $(-\delta, \delta) \subseteq X$ for every real number $\delta$ such that $0<\delta \leq 1$ .

It's seems in every calculus book I read that the choice of $\delta$ has always been $0<\delta \leq 1$ .

Could someone explain why $0<\delta \leq 1$ ?

**undefined** · July 17th 2010, 09:38 PM

Originally Posted by novice

If $f:X \rightarrow \mathbb{R}$ is defined by $f(x)=\frac{x}{x^2-1}$ , and we are interested in the behavior of $f$ near $0$ , then $1,-1 \not\in X$ . A natural choice of $X$ is $\mathbb{R}-\{-1,1\}$ , in which case $(-\delta, \delta) \subseteq X$ for every real number $\delta$ such that $0<\delta \leq 1$ .

It's seems in every calculus book I read that the choice of $\delta$ has always been $0<\delta \leq 1$ .

Could someone explain why $0<\delta \leq 1$ ?

Is there some more context that you haven't told us? Because in the context of epsilon-delta proofs, there's no such general restriction but just

"for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |ƒ(x) − L| < ε."

Obviously if you're considering open balls of radius delta centered at a point, then you need those to be defined, so you may have to choose a restriction accordingly.

**mohammadfawaz** · July 18th 2010, 05:00 AM

Originally Posted by novice

It's seems in every calculus book I read that the choice of $\delta$ has always been $0<\delta \leq 1$ .
Could someone explain why $0<\delta \leq 1$ ?

This is not a valid restriction on $\delta$ . The fact that you're only finding values between 0 and 1 is just a coincidence. If you need to choose a value for $\delta$ , you usually choose it in a way to serve your method for solving some problem. In fact, the only property about $\delta$ is that it can goes as small as you need without touching zero. but if the value 1000 is more suitable for you that 0.5, then you can choose 1000!

**novice** · July 18th 2010, 05:03 AM

Originally Posted by undefined

Is there some more context that you haven't told us? Because in the context of epsilon-delta proofs, there's no such general restriction but just

"for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |ƒ(x) − L| < ε."

Obviously if you're considering open balls of radius delta centered at a point, then you need those to be defined, so you may have to choose a restriction accordingly.

No information is left out.

The statement emphasizes that "if we are interested in the behavior of $f$ near $0$ where $f(x)=\frac{x}{x^2-1}$ ", and it closes by defining $0\leq \delta \leq 1$ . It's not about $f$ having a limit $L$ .

I understand that the deleted neighborhood of a point, say $a$ , is an opened interval $(a-\delta,a) \cup (a,a+\delta)$ , but why $0\leq \delta \leq 1$ ?

**mohammadfawaz** · July 18th 2010, 05:11 AM

In this example, since you need the limit near zero, the interval you should take is a neighborhood of zero as you said in the last sentence. But what are the allowable neighborhood of zero for this function? These are only the intervals of the form $(-\delta,0) \cup (0,+\delta)$ with $0< \delta < 1$ . If you ignore this restriction, then $\delta$ can take the value 1.5 (for example...anything larger than 1) and in this case you will get the interval $(-1.5,0) \cup (0,+1.5)$ on which the function is not defined (since it is not defined on 1 or -1). The whole idea is to cover all neighborhoods of zero on which f is safely defined.

**HallsofIvy** · July 18th 2010, 05:35 AM

In this particular case, if you are looking at the "behavior" of $\frac{x}{x^2- 1}= \frac{x}{(x- 1)(x+ 1)}$ for x close to 0 you want to avoid making the denominators 0 which happens when x= 1 or x= -1. That is why you will want to be sure that |x|< 1.

**novice** · July 18th 2010, 05:44 AM

Originally Posted by mohammadfawaz

In this example, since you need the limit near zero, the interval you should take is a neighborhood of zero as you said in the last sentence. But what are the allowable neighborhood of zero for this function? These are only the intervals of the form $(-\delta,0) \cup (0,+\delta)$ with $0< \delta < 1$ . If you ignore this restriction, then $\delta$ can take the value 1.5 (for example...anything larger than 1) and in this case you will get the interval $(-1.5,0) \cup (0,+1.5)$ on which the function is not defined (since it is not defined on 1 or -1). The whole idea is to cover all neighborhoods of zero on which f is safely defined.

In the given example we know that $x \not = 1 \wedge x \not = -1$ , now I can see why $0<\delta <1$ , but I saw in many books it seems arbitrarily the choice of $\delta$ has always been $0<\delta<0$ . I have a hunch that since $\delta$ is so small that it stand to reason not to have $\delta >1$ , but no one who wrote math books want to bother to clear the myth, or perhaps it's just too obvious to math savvy.

**novice** · July 18th 2010, 05:45 AM

Originally Posted by HallsofIvy

In this particular case, if you are looking at the "behavior" of $\frac{x}{x^2- 1}= \frac{x}{(x- 1)(x+ 1)}$ for x close to 0 you want to avoid making the denominators 0 which happens when x= 1 or x= -1. That is why you will want to be sure that |x|< 1.

You hit the mark.

**mohammadfawaz** · July 18th 2010, 05:50 AM

as for what I know, $\delta$ is a positive real number. That's how we define it. I never saw a general restriction on it (unless imposed by the problem as in you case). But as u said, more than 99% of the times, the value of $\delta$ is practically very small. Probably much smaller than 1 because we usually talk about limits and the we always say: "limit near the point a" (for example), hence, we don't expect to be more than 1 unit away from a. Just a common sense. But mathematically, nothing prevents $\delta$ from being large.

**novice** · July 18th 2010, 06:28 AM

Originally Posted by mohammadfawaz

But mathematically, nothing prevents $\delta$ from being large.

Agree. Nothing prevents it from being larger than 1, but it might not even be necessary to be any larger. Take $f(x) = \frac{|x|}{x}$ in the neigborhood of $0$ . The choice of $\delta$ is still $0<\delta <1.$

In this case, we know that $0<\delta <1$ is rather an exaggeration, and perhaps, it is just a standard treatment.

**theodds** · July 18th 2010, 08:35 AM

It should be noted that, if you find a $\delta$ that works, then any $\gamma < \delta$ must, by the nature the definition of a limit, also work. So, while it doesn't really matter substance wise, there's nothing wrong with requiring $\delta \in (0, 1)$ , and you could always accommodate this requirement by finding $\delta^* \in \mathbb{R}$ that works for the problem at hand, and just define $\delta = \min\{\delta^*, 1/2\}$ .

It's often useful to go ahead and just let $\delta$ or $\epsilon$ be less than a particular constant to facilitate limit finding. There's nothing wrong with choosing $\delta$ to be arbitrarily small, or assuming WLOG that $\epsilon$ is arbitrarily small.

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