Introduction to the Limit of a Function: Learn It 3

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Introduction to the Limit of a Function: Learn It 3

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point.

To see this, we now revisit the function $g(x)=\frac{|x-2|}{(x-2)}$ introduced at the beginning of the section (Figure 1 part b).

"Three 2 and x= -1 for x < 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 / (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two." width="975" height="434" — Figure 1. These graphs show the behavior of three different functions around $x=2$ .

As we pick values of $x$ close to $2$ , $g(x)$ does not approach a single value, so the limit as $x$ approaches $2$ does not exist—that is, $\underset{x\to 2}{\lim}g(x)$ DNE.

However, this statement alone does not give us a complete picture of the behavior of the function around the $x$ -value $2$ . To provide a more accurate description, we introduce the idea of a one-sided limit.

For all values to the left of $2$ (or the negative side of $2$ ), $g(x)=-1$ . Thus, as $x$ approaches $2$ from the left, $g(x)$ approaches $−1$ .

Mathematically, we say that the limit as $x$ approaches $2$ from the left is $−1$ . Symbolically, we express this idea as

lim_{x \to 2^{-}} g (x) = - 1

$\underset{x\to 2^-}{\lim}g(x)=-1$

Similarly, as $x$ approaches $2$ from the right (or from the positive side), $g(x)$ approaches $1$ . Symbolically, we express this idea as

lim_{x \to 2^{+}} g (x) = 1

$\underset{x\to 2^+}{\lim}g(x)=1$

one-sided limits

One-sided limits are limits approached from one direction—either from the left or the right.

Left-Sided Limit: For a function $f(x)$ on an interval ending at $a$ , if $f(x)$ approaches a specific value $L$ as the values of $x$ approaches $a$ from the left ( $x < a$ ), we denote this limit as:
$\underset{x\to a^-}{\lim}f(x)=L$
Right-Sided Limit: For a function $f(x)$ on an interval ending at $a$ , if $f(x)$ approaches a specific value $L$ as the values of $x$ approaches $a$ from the right ( $x > a$ ), we express this limit as:
$\underset{x\to a^+}{\lim}f(x)=L$

For the function $f(x)=\begin{cases} x+1, & \text{ if } \, x < 2 \\ x^2-4, & \text{ if } \, x \ge 2 \end{cases}$ , evaluate each of the following limits.

$\underset{x\to 2^-}{\lim}f(x)$
$\underset{x\to 2^+}{\lim}f(x)$

We can use tables of functional values again. Observe that for values of $x$ less than $2$ , we use $f(x)=x+1$ and for values of $x$ greater than $2$ , we use $f(x)=x^2-4$ .

Table of Functional Values for $f(x)=\begin{cases} x+1, & \text{ if } \, x < 2 \\ x^2-4, & \text{ if } \, x \ge 2 \end{cases}$
$x$	$f(x)=x+1$	$x$	$f(x)=x^2-4$
$1.9$	$2.9$	$2.1$	$0.41$
$1.99$	$2.99$	$2.01$	$0.0401$
$1.999$	$2.999$	$2.001$	$0.004001$
$1.9999$	$2.9999$	$2.0001$	$0.00040001$
$1.99999$	$2.99999$	$2.00001$	$0.0000400001$

Based on this table, we can conclude that

$\underset{x\to 2^-}{\lim}f(x)=3$
$\underset{x\to 2^+}{\lim}f(x)=0$ .

Therefore, the (two-sided) limit of $f(x)$ does not exist at $x=2$ . Figure 7 shows a graph of $f(x)$ and reinforces our conclusion about these limits.

"The

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Two-Sided Limits

To fully grasp how limits function, it’s essential to understand the connection between one-sided and two-sided limits.

A two-sided limit at a point exists only if the one-sided limits from both the left and the right converge to the same value. If there’s a discrepancy between the left and the right limits, the two-sided limit at that point does not exist.

two-sided limits

For a function $f(x)$ , defined over an interval including $a$ (except possibly at $a$ itself), we say the two-sided limit exists as $x$ approaches $a$ and equals $L$ if, and only if, both one-sided limits as $x$ approaches $a$ also equals $L$ .

lim_{x \to a} f (x) = L

$\underset{x\to a}{\lim}f(x)=L$ , if and only if

lim_{x \to a^{-}} f (x) = L

$\underset{x\to a^-}{\lim}f(x)=L$ and

lim_{x \to a^{+}} f (x) = L

$\underset{x\to a^+}{\lim}f(x)=L$