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## Calculus 1

### Unit 1: Lesson 4

Formal definition of limits (epsilon-delta)

# Formal definition of limits Part 3: the definition

The epsilon-delta definition of limits says that the limit of f(x) at x=c is L if for any ε>0 there's a δ>0 such that if the distance of x from c is less than δ, then the distance of f(x) from L is less than ε. This is a formulation of the intuitive notion that we can get as close as we want to L. Created by Sal Khan.

## Want to join the conversation?

• • Why are epsilon and delta must be always GREATER than 0? Why can't both of it be always GREATER or EQUAL to 0? •  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.
• Is it true if I say that f(c+δ) = L+ε and f(c-δ) = L - ε , for any function ? • Why do we use epsilon and delta and not some other greek letter? Is there a specific reason for using these letters? • 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" ? • 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.
• 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? • • 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.
• 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=∞? • Why is the statement "if |x-c|<δ, then |f(x)-L|<ε" instead "if |x-c|≤δ, then |f(x)-L|≤ε?" It is a tiny question but I need to know why there is a less than sign and not a less than or equal to sign.  