Definition 1.2.1 is a precise definition of the limit of the real-valued function $f\left(x\right)$. The limit is a number designated by the letter $L$. The notation for the "limit of $f\left(x\right)$ as $x$ approaches $a$" is

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

If this limit is the number $L$, write

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

As Definition 1.2.1 is read, keep in mind that all it is trying to say is that for $x$ "near" $a$, the values of $f\left(x\right)$ are "near" $L$. But also keep in mind that it took nearly 100 years for the mathematics community to settle on this form of the statement of what "near" should mean.

This definition rigorously captures the essence of "nearness." It says that if the values of $f\left(x\right)$ get arbitrarily close to the number $L$ when $x$ is sufficiently close to $a\&comma;$then $L$ is the limit to those function values. The nearness to $L$ of the values $f\left(x\right)$ is measured by $\left|f\left(x\right)-L\right|$ while the nearness of $x$ to $a$ is measured by $\left|x-a\right|$. The differences $\left|f\left(x\right)-L\right|$ must be smaller than ε, where ε represents any possible small number. These differences must be smaller than ε for all $x$ close enough to $x\&equals;a$. The nearness of $x$ to $a$ is dictated by the inequality $\left|x-a\right|<\mathrm{\delta }\&period;$

Because this definition might be difficult to fathom at first, several examples that interpret this definition graphically are provided.

The precise definition of a limit expressed in Definition 1.2.1 took mathematicians more than a century to formulate. The central idea expressed by it is easier to grasp if it can be seen graphically. The EpsilonDelta maplet is an interactive tool for visualizing the roles of $\mathrm{\&varepsilon;}$ and $\mathrm{\&delta;}$ in Definition 1.2.1. Its use is illustrated in Examples 1.2.1-4.$$

Definition 1.2.1 can be used to prove or disprove that a given number $L$ is, or is not, the limit. It cannot be used to find the value of $L$. Methods for finding $L$ appear in the next few sections. In each of  Examples 1.2.6-9, an animation captures the relationship between the intervals $\left|x-a\right|<\mathrm{\&delta;}$ and $\left|f\left(x\right)-L\right|<\mathrm{\&varepsilon;}$. Then, the following "algorithm" is applied to find $\mathrm{\&delta;}\left(\mathrm{\&varepsilon;}\right)$, and to verify that it satisfies Definition 1.2.1.

Note: For a function that decreases in the vicinity of $x\&equals;a$, the two equations for ${\mathrm{\&delta;}}_L$ and ${\mathrm{\&delta;}}_R$ would respectively be

$f\left(a\&plus;{\mathrm{\&delta;}}_R\right)\&equals;L-\mathrm{\&varepsilon;}$ and $f\left(a-{\mathrm{\&delta;}}_L\right)\&equals;L\&plus;\mathrm{\&varepsilon;}$ 

The use of Definition 1.2.1 to verify a limit is illustrated in Examples 1.2.5-8.

For two-sided limits, values of the independent variable on both sides of the limit point must be considered (assuming they are in the domain of the function). One-sided limits only consider values of the independent variable on one side of the limit point. These differences are summarized in Table 1.2.4.

Definition 1.2.1 can be applied to a one-sided limit with the same technique used in Examples 1.2.6 - 9. However,  the equation determining $\mathrm{\&delta;}\left(\mathrm{\&varepsilon;}\right)$ must be selected from Table 1.2.5, which takes into account how $x\&equals;a$ is approached, and whether the function is increasing or decreasing in a neighborhood of $x\&equals;a$.

Table 1.2.3 summarizes some of the main points of the section.