The Limit Concept

Disclaimer. The following can be found in numerous standard texts on Calculus, such as: Suppose $f$ is a real-valued function of $x$ and $a$ is a real number. The expression: $$ \lim_{x\rightarrow a} f(x) = L $$ means that $f(x)$ can be made as close to $L$ as desired, by making $x$ sufficiently close to $a$, but without actually letting $x$ be $a$. In this case, we say that "the limit of $f(x)$, as $x$ approaches $a$, is $L$".
The exact definition is as follows. Let $f(x)$ be a function defined on an interval that contains $x = a$ . Then we say: $$ \lim_{x\rightarrow a} f(x) = L $$ if for every number $\epsilon > 0$ there is some number $\delta > 0$ such that: $$ | f(x) - L | < \epsilon \quad \mbox{whenever} \quad 0 < | x - a | < \delta $$ Note that $f(x)$ is possibly not defined at $x = a$ .
As an example, consider the following limit: $$ \lim_{x\rightarrow 1} \frac{x^2-1}{x-1} = 2 $$ Indeed. Because $a = 1$ , but $0 < |x-a|$ , so $x \neq 1$ , we may safely write: $$ \lim_{x\rightarrow 1} \frac{x^2-1}{x-1} = \lim_{x\rightarrow 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x\rightarrow 1} (x+1) $$ Now take $\delta = \epsilon$, then there always exists some number $\delta$, namely $\epsilon$, such that $|(x+1)-2| = |x-1| < \epsilon$ whenever $|x-1| < \delta$.
Closely related to the definition of a limit, but not identical to it, is the definition of continuity. A function $f$ is continuous at a number $a$ if: $$ \lim_{x\rightarrow a} f(x) = f(a) $$ Example. Let $f(x)$ be defined by: $$ f(x) = \left\{ \begin{array}{ll} 2 & \mbox{for} \quad x = 1 \\ (x^2-1)/(x-1) & \mbox{for} \quad x \neq 1 \end{array} \right. $$ Then we have proved that $f(x)$ is continuous at $x=1$. If we define instead: $$ f(x) = \left\{ \begin{array}{ll} 0 & \mbox{for} \quad x = 1 \\ (x^2-1)/(x-1) & \mbox{for} \quad x \neq 1 \end{array} \right. $$ Then $f(x)$ is not continuous at $x=1$, because: $\lim_{x\rightarrow 1} f(x) = 2 \neq 0$.
Suppose, again, that $f$ is a real-valued function of $x$. The expression: $$ \lim_{x \rightarrow \infty} f(x) = L $$ means that $f(x)$ can be made as close to $L$ as desired, by making $x$ large enough. Without actually letting $x$ be $\infty$ would be a trivial addendum in this case. We say that "the limit of $f(x)$, as $x$ approaches infinity, is $L$". The exact definition is as follows. Let $f(x)$ be a function defined for sufficiently large values of $x$. Then we say: $$ \lim_{x\rightarrow \infty} f(x) = L $$ if for every number $\epsilon > 0$ there is some number $N > 0$ such that: $$ | f(x) - L | < \epsilon \quad \mbox{whenever} \quad x > N $$ As an example, consider the following limit: $$ \lim_{x\rightarrow \infty} 1/x = 0 $$ We find that this limit is equal to zero. Indeed: $$ | 1/x - 0 | < \epsilon \quad \mbox{whenever} \quad x > N \quad \mbox{with} \quad N = 1/\epsilon $$ Last but not least we have infinite limits. Symbolically: $$ lim_{x\rightarrow \infty} f(x) = \infty $$ if for every number $M > 0$ there is some number $N > 0$ such that: $$ f(x) > M \quad \mbox{whenever} \quad x > N $$ But, in some circles, such infinite limits are simply said not to exist.
Claimer. The following will not be found in any standard texbook on calculus. Let's restrict attention to the first limit definition given above: $$ \lim_{x\rightarrow a} f(x) = L $$ if for every number $\epsilon > 0$ there is some number $\delta > 0$ such that: $$ | f(x) - L | < \epsilon \quad \mbox{whenever} \quad 0 < | x - a | < \delta $$ Numbers like $\epsilon > 0$ and $\delta > 0$ are known in computational work as errors. Even the real numbers themselves in a digital computer are not error free; as is obvious if one seeks to represent irrational numbers such as $\pi$ or $\sqrt{2}$. If $x$ is the floating point number representing $\pi$ and $\delta$ is the "machine eps" (i.e. an error) then only the following is true: $$ x \not \equiv \pi \quad \mbox{and} \quad 0 < | x - \pi | < \delta $$ Even worse. While carrying out computations with real machine numbers, errors tend to accumulate. This is actually reflected within the $(\delta,\epsilon)$ formalism for limits. Let, for example, $|x-a| < \delta_x$ and $|y-a| < \delta_y$, where $(a,b)$ are supposed to be "exact" and $(x,y)$ are machine numbers. Next calculate: $$ | (x+y) - (a+b) | = | (x-a) + (y-b) | \le |x-a| + |y-b| < \delta_x + \delta_y $$ So the error in the sum $(x+y)$ is most probably greater than one of the errors $(\delta_x,\delta_y)$ , namely the sum of these. Similar expressions can easily be derived for the other elementary operations. And are encountered as well in common proofs involving limits. We may conclude that the concept of a limit actually represents error processing - though in an idealized manner. To put it the other way around: the materialization of the limit concept is error processing.
Now take another look at the limit definition, especially the clause "if for every number $\epsilon > 0$ there is some number $\delta > 0$ such that". If $f(x)$ is interpreted as (the end result of) a calculation, then it simply says that the error propagation of that calculation, though dependent upon initial errors $\delta$, must be guaranteed to be limited within a certain pre-defined bound $\epsilon$.