Intuitive Idea of Limit

Before limits can be defined precisely, it helps to build the idea of a limit

informally -- by asking what value a function $f(x)$ gets closer and closer to as $x$ itself

gets closer and closer to some fixed number $a$, without $x$ ever actually reaching $a$.

A motivating example. Consider $f(x) = \dfrac{x^2-1}{x-1}$. This function is not defined

at $x=1$, since substituting $x=1$ gives $\dfrac{0}{0}$, a meaningless expression. Yet nothing

stops $x$ from taking values very close to $1$, on either side of it. The table below records

$f(x)$ for a sequence of such values.

As $x$ approaches $1$ from below (values less than $1$) or from above (values greater than

$1$), $f(x)$ approaches $2$ from both sides -- even though $f$ itself has no value AT $x=1$.

This number $2$ that $f(x)$ approaches is called the limit of $f(x)$ as $x$ tends to $1$,

written

(Algebraically this is easy to confirm: for $x\neq1$, $f(x)=\dfrac{(x-1)(x+1)}{x-1}=x+1$, which

does tend to $2$ as $x\to1$ -- Section 3 makes this cancellation technique precise.)

Left-hand and right-hand limits. The values of $x$ approaching $a$ from below (i.e.

$x<a$) give the left-hand limit, written $\displaystyle\lim_{x\to a^-}f(x)$; the values

approaching from above ($x>a$) give the right-hand limit, written

$\displaystyle\lim_{x\to a^+}f(x)$.

Definition (existence of a limit). The limit $\displaystyle\lim_{x\to a}f(x)$ is said to

exist and equal $L$ if and only if both one-sided limits exist and are equal to the same

value $L$:

If the left-hand and right-hand limits are unequal (or either one fails to exist), the limit

$\displaystyle\lim_{x\to a}f(x)$ itself does not exist -- this is the single most common way a

limit problem "goes wrong", and checking both one-sided limits separately is the standard first

step whenever a function changes its formula at $x=a$ (a piecewise function) or involves an

expression like $|x|$ that behaves differently on either side of a point.

Why the value at $a$ itself is irrelevant. The definition of $\displaystyle\lim_{x\to a}f(x)$

never asks what $f(a)$ actually is -- only what $f(x)$ approaches as $x$ gets arbitrarily close

to $a$ while remaining different from $a$. This is why the limit of $\dfrac{x^2-1}{x-1}$ at

$x=1$ exists and equals $2$ even though the function itself is undefined there; it is also why a

function can later fail to be continuous at a point even though its limit there exists (a

topic outside this chapter's scope, but the distinction begins exactly here).