Question 1

why is epsilon always greater than delta?

Accepted Answer

It's not! Sometimes epsilon is greater than delta, sometimes delta is greater than epsilon.

Question 2

Why are epsilon and delta must be always GREATER than 0? Why can't both of it be always GREATER or EQUAL to 0?

Accepted Answer

Because limits have to do with numbers being within a certain (positive) distance of other numbers, and epsilon and delta are (positive) distances. If the distance between two numbers is zero, the numbers are equal, and so they are trivially within any positive distance of each other, but that is not what the idea of limits is about.

Question 3

Is it true if I say that  f(c+δ) = L+ε and f(c-δ) = L - ε , for any function ?

Accepted Answer

No it is not true. Assume we have a function f(x)=x^2. Obviously, the limit as x->0 is 0, so L=0. We can't have any negative y values, in this function, so there is no such thing as L-ε, because that would be negative.

Question 4

Why do we use epsilon and delta and not some other greek letter? Is there a specific reason for using these letters?

Accepted Answer

The choice of these letters is arbitrary, but there's a long tradition of using these letters for this purpose. In particular, epsilon is typically used whenever referring to an arbitrarily small amount. The famous mathematician Paul Erdos extended the concept, humorously referring to small children as epsilons.

Question 5

Why does he use a < sign rather than a less than or equal to sign to say that x is within delta of c or to say that f(x) is within epsilon of L? Wouldn't the x value at the point (c+or-delta,f(c+or-delta)) be within the required range?

Accepted Answer

He uses the < sign because he isn't trying to find the limit, he is just trying to get close to the limit.

Question 6

How can we define one-sided limits?
By using "x-c < -d and x < c" or "x-c < d and x > c" instead of "|x-c| < d" ?

Accepted Answer

I will not formulate the most general way of defining one-sided limits (it requires some knowledge of point-set topology), but suppose ƒ is a real-valued function defined on a set containing an open interval of the form (a, b), where a < b are two real numbers.

I) One says that ƒ has right-sided limit L at a if and only if there for every real number ε > 0 exists a real number δ > 0 such that |ƒ(x) - L| < ε for all real numbers x such that a < x < a + δ. (This latter condition on x may be rephrased as 0 < x - a < δ.) In other words, the condition |ƒ(x) - L| < ε is to hold for all x in the interval (a, a + δ) for some δ > 0.

II) One says that ƒ has left-sided limit L at b if and only if there for every real number ε > 0 exists a real number δ > 0 such that |ƒ(x) - L| < ε for all real numbers x such that b - δ < x < b. (This latter condition on x may be rephrased as -δ < x - b < 0, i.e., 0 < b - x < δ.) In other words, the condition |ƒ(x) - L| < ε is to hold for all x in the interval (b - δ, b) for some δ > 0.

There is a more general notion of one-sided limits. If ƒ is defined on a set X of real numbers, and if p is a limit point of the intersection of X with (p, +∞), we say that ƒ has right-sided limit L at p if and only if for all ε > 0 there exists δ > 0 such that |ƒ(x) - L| < ε for all x in X with p < x < p + δ. One defines left-sided limits similarly.

Question 7

Can't there be multiple deltas for an epsilon?

Accepted Answer

Let `0 < δ' < δ` be real numbers, and let `x` and `c` be real numbers. Observe that `|x - c| < δ'` implies that `|x - c| < δ`. Hence `0 < |x - c| < δ'` implies `0 < |x - c| < δ`. Therefore, if you have found one `δ` for which the condition holds, then any other `δ'` less than `δ` also works.

Question 8

The definition of limits provided assumes that f(x) is defined for all real numbers, but if f(x) is not defined for all real numbers, then ε cannot be any number you want which is greater than zero. If f(x) is not defined on f(c) itself, it's not a problem since x cannot equal c from definition 0 < |x - c| < δ anyway. But perhaps that isn't an issue in practice, since when f(x) is undefined, | f(x) - L | < ε is also undefined, and you can just ignore it? Any clarity as to how this definition can be used in practice would be appreciated. Am I right in thinking this definition does not apply when c=∞ or when L=∞?

Accepted Answer

The definition doesn't apply when c or L are infinite because we demand that f(x) have real inputs and outputs, and infinity isn't a real number.

𝜀 can be chosen completely arbitrarily no matter where f(x) is defined. If 𝜀 is very large, larger than the range of f(x), then |f(x)-L| will just be much smaller than 𝜀. So what?

Question 9

i understand why we need limits but what is the point of the epsilon delta definition

Accepted Answer

Newton could not rigorously prove any of his Calculus despite a lifetime of effort. The way continuous functions are introduced to beginning calculus students is to ask something intuitive like: "As x gets this close to c, how close does f(x) get to f(c)?” This hand-wavy approach gives students an intuitive feel for the topic, but is not useful for rigorous proofs.  Many decades after Newton's struggles, the Bohemian  mathematician Bolzano turned the question around by asking essentially: "If we want f(x) to get this close to f(c), how close does x have get to c?" This is the basis of the epsilon-delta definition, which finally allowed mathematicians to rigorously prove the calculus invented by Newton (and Leibnitz). It is counter-intuitive to math students at first, but after a while they internalize it and it becomes second nature. Remarkably, over such a long time the smartest mathematicians (including Newton) missed discovering that crucial one-liner definition. Analysis (of which calculus is a part) involves a lot of proving that something is within epsilon of something else.  As a footnote, during my school days, another student remarked to me: "Bruce came within epsilon of flunking out last year".

Calculus, all content (2017 edition)

Course: Calculus, all content (2017 edition) > Unit 1

Formal definition of limits Part 3: the definition

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