Unizor - Derivatives - Function Limit Definition

Notes to a video lecture on http://www.unizor.com


Function Limit -
Two Definition

A sequence {X_n} from the functional viewpoint can be considered as a function from the set of all natural numbers N(domain of this function) into a set of all real numbers R (co-domain of this function).

Order number n is an argumentof this function, while X_nrepresents a value of this function for argument n.

When we consider a limit of asequence, we have in mind a process of increasing the argument n without any bounds (to infinity, as we might say) and observing the convergence or not convergence of the values X_n to some real number, which in case of convergence is called the limit of this sequenceas order number n increases toinfinity.

More rigorously, real numbera is a limit of a sequence {X_n}, if for any (however small) ε > 0 there exists order number Nsuch that
|a - X_n| ≤ ε for any n ≥ N.

In this lecture we will expand our field from sequences to functions of any real argument and real values. We will generalize the concept of a limit to a process of an argument not only increasing to infinity, but converging to any real number.
After this we will be ready to define derivatives and other interesting principles of Calculus.

To consider a function instead of a sequence presents no efforts, since a sequence is a function. All we do is expanding a domain from all natural numbers N to all real numbers R.

Without much efforts we can define a limit of a function as its argument increases to infinity. This practically repeats the definition of a limit of a sequence and looks like this.

Real number a is a limit of function f(x) when x increases to infinity, if for any positive distance ε, however small, there is a real number r such that for all x ≥ r it is true thatf(x) is closer to a than distance ε, that is |f(x)−a| ≤ ε.

Let's express this symbolically.
∀ ε>0 ∃ r: x ≥ r ⇒ |f(x)−a| ≤ ε

Having a set of natural numbersN as a domain dictates only one type of process where we can observe the change of sequence values - when order number nincreases to infinity.
With a domain expanded to all real numbers we have more choices. One such way, as defined above, is to increase an argument of a function to infinity - total analogy with sequences. Another is to decrease the argument to negative infinity.
Here is how it can be defined.

Real number a is a limit of function f(x) when x decreases to negative infinity, if for any positive distance ε, however small, there is a real number rsuch that for all x ≤ r it is true that f(x) is closer to a than distance ε, that is |f(x)−a| ≤ ε.

Let's express this symbolically.
∀ ε>0 ∃ r: x ≤ r ⇒ |f(x)−a| ≤ ε

Our final expansion of a limit of sequence to limit of function is to define a limit of function when its argument gets closer and closer (converges) to some real number instead of going to positive or negative infinity.

First of all, we have to define this process of convergence of an argument to some real number more precisely.
Obvious choice is to measure the distance between an argument x and some real number r our argument, supposedly, approaches. So, if xis changing from x₁ to x₂, tox₃... to x_n etc. such that sequence {|x_n−r|} isinfinitesimal (that is converges to zero), we can say that xconverges to r.

It is very important to understand that argument x can converge to value r in many different ways forming different infinitesimals. For example, sequence x_n = r+1/n is one such way. Another is x_n = r·(1+1/n). Yet another is x_n = r·2^1/n.
Even more sophisticated way to converge is for x to approach ronly on rational numbers skipping irrationals or, inversely, only on irrational numbers, skipping rationals.
As you see, convergence of an argument to a specific value can be arranged in many different ways, but in any way the distance |x−r| must be infinitesimal, that is must converge to zero.

Let's examine now the behavior of a function f(x) as its argument x converges to valuer. It is natural to assume that function f(x) converges to valuea when its argument xconverges to value r if |f(x)−a|is an infinitesimal when |x−r| is infinitesimal.
Symbolically, we can describe this as
{x_n}→r ⇒ {f(x_n)}→a

It is very important to understand that there might be cases when x converges to r in some way (that is, sequence{|x_n−r|} is infinitesimal) and corresponding sequence of function values f(x_n) converges to a, but, if argument xconverges to r in some other way, function values f(x_n) do not converge to a.
Here is an interesting example. Consider a function f(x) that takes value 0 for all rational arguments x and takes value 1for all irrational x. Now let xapproach point r=0 stepping only on rational numbers likex_n=1/n. All values of f(x_n) will be 0 and we could say that f(x)converges to 0 as x converges to 0. But if we step only on irrational numbers like x_n=π/n, our function will take valuesf(x_n) equaled to 1 and we would assume that f(x) converges to 1as x converges to 0. This does not seem right.

It is appropriate then to formulate the concept of limit in terms of sequences as follows.
Value a is a limit of functionf(x) when its argument xconverges to real number r, if for ANY sequence of argument values {x_n} converging to r the sequence of function values {f(x_n)} converges to a.

Symbolically:
∀{x_n}→r: {f(x_n}→a

Though logically we have come up with a correct definition of a limit of function when its argument converges to a specific real number, it is not easy to verify that concrete function has a concrete limit when its argument converges to concrete real number. We cannot possibly examine ALL the ways an argument approaches its target.
Let's come up with another (equivalent) definition of a limit that can be used to constructively prove statements about limits.

Again, we will use analogy with sequence limits.
The key point to a definition of a limit for a sequence was that, when order number n is sufficiently large (non-mathematically, we can say "sufficiently close to infinity"), the values of sequence members are sufficiently close to its assumed limit. The degree of order number to be "sufficiently close to infinity" (that is, sufficiently large, greater than some number) depends on how close we want our sequence to be to its limit. Greater closeness of a sequence to its limit necessitates larger order number, that is its greater "closeness to infinity", so to speak.

For function limits we will approach more constructive definition analogously. If we assume that some real number ais a limit of a function f(x) as xconverges to r, then for any degree of closeness between a function and its limit there should be a neighborhood of value r in which (that is, if x is within this neighborhood) this degree of closeness between a function and its limit is observed. Greater closeness requires a narrower neighborhood.

Expressing it more precisely, for any positive ε there should be positive δ such that, if x is within δ-neighborhood of r(that is, |x−r| ≤ δ), then f(x) will be within ε-neighborhood of a(that is, |f(x)−a| ≤ ε).

Symbolically:
∀ ε>0 ∃ δ>0:
|x−r| ≤ δ ⇒ |f(x)−a| ≤ ε

Sometimes the last two inequalities in the above definition are specified as "less" instead of "less or equal". It makes no difference.