## Limits

**First, what’s the problem?**

Assume we want to calculate the derivative of the function , say, at the point . This derivative is the number, defined using the divided difference when is “very small”. What does it mean “very small”? We cannot let be *exactly* zero, since division by zero is forbidden. On the other hand, if , then the above expression is never equal to 4 (as expected) *precisely*, so in any case the “derivative” cannot be 4, as we want. To resolve this controversy, Leibniz introduced mysterious “*differentials*” which disappear when added to usual numbers, but whose ratio has precise numerical meaning.

The approach by Leibniz can be worked out into a rigorous mathematical theory, called *nonstandard analysis*, but historically a different approach, based on the notion of **limit **(of sequence, function, …), prevailed.

### Limit of a sequence

Consider an (infinite) sequence of real numbers. We say that it stabilizes (מתיצבת) at the value , if only finitely many terms in this sequence can be different from , and the remaining infinite “tail” consists of the repeated value . Since among the finitely many numbers one can always choose the maximal one (denoted by ), we say that the sequence $latex\{a_n\}$ stabilizes, if

Obviously, stabilizing sequences are not interesting, but their obvious properties can be immediately listed:

- Changing any finite number of terms in a stabilizing sequence keeps it stabilizing and vice versa;
- If a sequence stabilizes at , and another sequence stabilizes at , then the sum-sequence stabilizes at , the product-sequence at $AB$.
- The fraction-sequence may be not defined, but if , then only finitely many terms can be zeros, just change them to nonzero numbers and then the fraction-sequence will be defined for all and stabilize at .
*Exercise*: formulate properties of stabilizing sequences, involving inequalities.

### Blurred vision

As we have introduced the real numbers, to test their equality requires to check infinitely many “digits”, which is unfeasible. All the way around, we can specify a given *precision* (it can be chosen rational). Then one can replace the genuine equality by -equality, saying that two numbers are -equal, if . This is a bad notion that will be used only temporarily, since it is not transitive (for the same fixed value of ). Yet as a temporary notion, it is very useful.

We say that an infinite sequence -stabilizes at a value for a given precision , if only finitely many terms in the sequence are not -equal to . Formally, this means that

### Spectacles can improve your vision

The choice of the precision is left open so far. In practice it may be set at the level which is determined by the imperfection of our measuring instruments, but since we strive for a mathematical definition, we should not set any artificial threshold.

**Definition.** A sequence of real numbers is said to *converge to the limit* , if *for any* given precision this sequence -stabilizes at . The logical formula for the corresponding statement is obtained by adding one more quantifier to the left:

If we want to claim that the sequence is converging without specifying what the limit is, one more quantifier is required:

Of course, this formula is inaccessible to anybody not specially prepared for the job, this is why so many students shuttered their heads over it.

### Obvious examples

- , or, more generally, . The limit is zero.
- . Joke.
- . Diverges.
- These rules (plus the obvious rules concerning the arithmetic operations) allow to decide the convergence of any sequence whose general term is a rational function of .
- Exceptional cases are very rare: e.g., when …

### Limits of functions

Let be a function defined on a subset , and a point *outside* the doomain of . We want to “extend” the function to this point if this makes sense.

For a given precision we say that is -constant on a set , if there exists a constant such that .

**Definition.** The function is said to have a limit equal to at a point , if

- all intersections between and small intervals are non-empty and
- such that the function restricted on is -constant.

In other words, the function is -indistinguishable from a constant on a sufficiently small open interval centered around .

**Remark.** One can encounter situations when the function is defined at some point , but if we delete this point from , then the function will have a limit $A$ at this point. If this limit coincides with the original value , then the function is well-behaved (we say that it is continuous). If the limit exists but is different from , then we understand that the function was intentionally twisted, and if we change its value at just one point, then it will become continuous. If has no limit at if restricted on , we say that $katex f$ is discontinuous$ at .

Clearly, such extension by taking limit is possible (if at all) only for points at “zero distance” from the domain of the function.

For more detail read the lecture notes from the past years.

### Series

As was mentioned, the problem of calculating limits of explicitly given (i.e., elementary) functions is usually not very difficult. The real fun begins when there is no explicit formula for the terms of the sequence (or the function). This may happen if the sequence is produced by some recurrent (inductive) rule.

The most simple case occurs where the rule is simply summation (should we say “correction”?) of a known nature:

.

If we denote the added value by , then the sequence will take the form

. If we can perform such summations explicitly and write down the answer as a function of , it would be great.

**Example.** Consider the case where . Then we get a “telescopic” sum which can be immediately computed. But this is rather an exception…

Another example is the geometric progression where for some constant .

In general we cannot write down the sum as a function of , which makes the task challenging.

**Definition.** Let be a real sequence. We say that the infinite series converges, if the sequence of its finite sums has a (finite) limit.

**Examples.**

- The geometric series converges if and diverges if .
- The harmonic series diverges.
- The inverse square series converges.

The last two statements follow from the comparison of the series with patently diverging or patently converging series (fill in the details!).

Later on we will concentrate specifically on the series of the form with a given sequence of “Taylor coefficients” which contain a parameter . Considered as the function of , these series exhibit fascinating properties which make them ubiquitous in math and applications.