2.2 The limit of a function  (Page 3/12)

We can make the following observations about these two limits.

1. For the first limit, observe that as x approaches a , so does $f\left(x\right),$ because $f\left(x\right)=x.$ Consequently, $\underset{x\to a}{\text{lim}}x=a.$
2. For the second limit, consider [link] .

Observe that for all values of x (regardless of whether they are approaching a ), the values $f\left(x\right)$ remain constant at c . We have no choice but to conclude $\underset{x\to a}{\text{lim}}c=c.$

The existence of a limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

Evaluating a limit that fails to exist

Evaluate $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ using a table of values.

[link] lists values for the function $\text{sin}(1\text{/}x)$ for the given values of x .

Table of functional values for $\underset{x\to 0}{\text{lim}}\text{sin}\left(\frac{1}{x}\right)$

x	$\text{sin}\left(\frac{1}{x}\right)$		x	$\text{sin}\left(\frac{1}{x}\right)$
−0.1	0.544021110889		0.1	−0.544021110889
−0.01	0.50636564111	0.01	−0.50636564111
−0.001	−0.8268795405312	0.001	0.826879540532
−0.0001	0.305614388888	0.0001	−0.305614388888
−0.00001	−0.035748797987	0.00001	0.035748797987
−0.000001	0.349993504187	0.000001	−0.349993504187

After examining the table of functional values, we can see that the y -values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x -values approaching 0:

$\frac{2}{\pi },\frac{2}{3\pi },\frac{2}{5\pi },\frac{2}{7\pi },\frac{2}{9\pi },\frac{2}{11\pi }\text{,….}$

The corresponding y -values are

$1,\mathrm{-1},1,\mathrm{-1},1,\mathrm{-1}\text{,….}$

At this point we can indeed conclude that $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write $\underset{x\to 0}{\text{lim}}\text{sin}\left(1\text{/}\mathit{\text{x}}\right)$ DNE.) The graph of $f\left(x\right)=\text{sin}\left(1\text{/}x\right)$ is shown in [link] and it gives a clearer picture of the behavior of $\text{sin}(1\text{/}x)$ as x approaches 0. You can see that $\text{sin}(1\text{/}\mathit{\text{x}})$ oscillates ever more wildly between −1 and 1 as x approaches 0.

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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 $g\left(x\right)=\left|x-2\right|\text{/}\left(x-2\right)$ introduced at the beginning of the section (see [link] (b)). As we pick values of x close to 2, $g\left(x\right)$ does not approach a single value, so the limit as x approaches 2 does not exist—that is, $\underset{x\to 2}{\text{lim}}g\left(x\right)$ 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), $g\left(x\right)=\mathrm{-1}.$ Thus, as x approaches 2 from the left, $g\left(x\right)$ approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as