2.2 The limit of a function  (Page 4/12)

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

We can now present an informal definition of one-sided limits.

Evaluating one-sided limits

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

1. $\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)$
2. $\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)$

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

Table of functional values for $f\left(x\right)=\{\begin{array}{l}x+1\text{if}x<2\\ x^2-4\text{if}x\ge 2\end{array}$

x	$f(x)=x+1$		x	$f(x)=x^2\mathrm{-4}$
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 a. $\underset{x\to 2^{-}}{\text{lim}}f\left(x\right)=3$ and b. $\underset{x\to 2^{+}}{\text{lim}}f\left(x\right)=0.$ Therefore, the (two-sided) limit of $f\left(x\right)$ does not exist at $x=2.$ [link] shows a graph of $f\left(x\right)$ and reinforces our conclusion about these limits.

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Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in [link] .

Infinite limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

We now turn our attention to $h\left(x\right)=1\text{/}{(x-2)}^2,$ the third and final function introduced at the beginning of this section (see [link] (c)). From its graph we see that as the values of x approach 2, the values of $h\left(x\right)=1\text{/}{(x-2)}^2$ become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of $h\left(x\right)$ as x approaches 2 is positive infinity. Symbolically, we express this idea as