$f\left(x\right)\=\left\{\begin{array}{cc}\frac{x^2-1}{x-1}& x\ne 1\\ \mathrm{undefined}& x\=1\end{array}\right.\=\left\{\begin{array}{cc}x\+1& x\ne 1\\ \mathrm{undefined}& x\=1\end{array}\right.$ 

In everyday speech, the word "limit" usually denotes some sort of boundary. In the calculus, the term is applied to functions, and in a much more precise, but related, sense. It is a "bounding" number to which a function's values get arbitrarily close. In particular, $L$ is the limit of the function $f\left(x\right)$ as $x$ approaches $a$, if the numbers $f\left(x\right)$ become increasingly close to $L$ when $x$ becomes increasingly close to $a$.

The formal way of writing this relationship between, $f\left(x\right)$, $a$, and $L$ is

 $\underset{x\to a}{lim}f\left(x\right)\=L$ 

The word "limit" is captured as an operator "lim" under which "$x\to a$" denotes "$x$ approaches $a$". This operator is applied to the function $f$, resulting in the number $L$. The notation means that the nearer the values of $x$ are to the number $a$, the closer the values of $f\left(x\right)$ are to the number $L$. If $L$ is the limit of $f\left(x\right)$ as $x$ approaches $a$, then the values of the function are very nearly the number $L$ for values of $x$ very nearly the number $a$, but the function itself need not even be defined at $x\=a$.

If the function is actually defined at $x\=a$ so that $f\left(a\right)$ is a valid number, that number may or may not be the value $L$! The limit at $x\=a$ merely expresses something about the value of the function near $x\=a$, not at $x\=a$.

This section views the limit naively, the way it was understood prior to the formal definition that evolved more than a century after it was first used in the calculus by Newton and Leibniz. The investigations and algebraic manipulations found in this section are based on the idea that "limit" means "$x$ near $a$ guarantees $f\left(x\right)$ near $L$."  

This section uses graphs to estimate a limit, and develops some basic manipulative techniques for finding the actual limiting value $L$. These techniques will be combined to develop the concepts of one-sided limits and to support an initial look at two special trigonometric limits. The section concludes with a short reminder that graphs and tables-of-value can be misleading.

Many limit calculations are illuminated by graphs and/or a table of values. However, it is easy to devise pathological functions for which such devices fail to provide a correct evaluation of a limit. Insofar as these devices are useful, their application is certainly encouraged. But the ultimate caution remains: only analytic methods are completely trustworthy. See Example 1.1.1 and Example 1.1.2.

The following examples will illustrate two basic techniques for evaluating the limit of an expression for which direct evaluation would generate a division-by-zero error. Hence, each method applies to fractions, the one to a rational function (the ratio of two polynomials) and the other to fractions that contain the square-root function. The first technique will be called factor and cancel (see Example 1.1.3); and the other, rationalize the numerator (see Example 1.1.4).

Later, Example 1.1.8 will illustrate Principle 1.1.1, another analytic principle for determining limits of certain functions that need not be fractions.

So, if $f$ is a product in which one factor goes to zero, but the other factor is merely bounded, then the limit of the product will be zero.

A piecewise-defined function is one whose domain is subdivided, and a different rule is used on each of these different subdivisions. For such a function, obtaining a limit at a point where the subdivisions meet requires obtaining the limits of the separate rules. Such limits are called one-sided limits. The function itself has a limit (the two-sided limit) at that point only if the two one-sided limits exist and have the same value.

For example, if $f\left(x\right)\&equals;\&lcub;\begin{array}{cc}g\left(x\right)& x<a\\ h\left(x\right)& x\ge a\end{array}$, then determining $\underset{x\to a}{lim}f\left(x\right)$ requires determining the two one-sided limits in Table 1.1.1.

The limit from the left (left-hand limit) of $x\&equals;a$ is taken through values of $x$ that are to the left of $a$, and this approach to $a$ is designated by the minus sign appended to the $a$. Of course, this limit is taken of the "left-hand" rule, that is, of the function $g\left(x\right)$.

Similarly, the limit from the right (right-hand limit) of $x\&equals;a$ is taken through values of $x$ that are to the right of $a$, and this approach to $a$ is designated by the plus sign appended to the $a$. Clearly, this limit is taken of the "right-hand" rule, that is, of the function $h\left(x\right)$. See Example 1.1.5 and Example 1.1.6.

Periodic functions such as the sine and cosine functions oscillate, and have especially interesting properties when composed with other functions. In Example 1.1.7 and Example 1.1.8, limits for such oscillatory functions are examined.

The number $\mathrm{\&pi;}$ is irrational because it cannot be represented as the ratio of two integers. Moreover, it is transcendental, that is, it is not the zero of a non-constant polynomial equation with rational coefficients.

The related limit $e^x\&equals;\underset{h\to 0}{lim}{\left(1\&plus;h x\right)}^{1\&sol;h}$ defines the exponential function whose base is the "exponential e." To see this, write

where $h x\&equals;H$, and $H\to 0$ as $h\to 0$.

Implement this number in Maple by selecting the symbol $\&ExponentialE;$ from the Common Symbols palette or from the Constants and Symbols palette. It can also be obtained by typing an ordinary e in Math mode, and invoking Command Completion either from the keyboard (Escape key, or Control + SpaceBar) or from the Tools menu. In the pop-up generated by Command Completion, select Exponential 'e'. Typing as text an ordinary "e" that simply looks like the exponential e is not sufficient. The exponential e, which in Maple is actually the number $\exp \left(1\right)$, must be inserted by one of the devices delineated above.

Table 1.1.3 summarizes some of the main points of this section.