A Graphical Approach To Precalculus With Limits Pdf

[Ashley]

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Figure 1 provides a visual representation of the mathematical concept of limit. As the input value [latex]x[/latex] approaches [latex]a[/latex], the output value [latex]f\left(x\right)[/latex] approaches [latex]L[/latex].

This notation indicates that as [latex]x[/latex] approaches [latex]a[/latex] both from the left of [latex]x=a[/latex] and the right of [latex]x=a[/latex], the output value approaches [latex]L[/latex].

Notice that [latex]x[/latex] cannot be 7, or we would be dividing by 0, so 7 is not in the domain of the original function. In order to avoid changing the function when we simplify, we set the same condition, [latex]x\ne 7[/latex], for the simplified function. We can represent the function graphically as shown in Figure 2.

Notice that the limit of a function can exist even when [latex]f\left(x\right)[/latex] is not defined at [latex]x=a[/latex]. Much of our subsequent work will be determining limits of functions as [latex]x[/latex] nears [latex]a[/latex], even though the output at [latex]x=a[/latex] does not exist.

A quantity [latex]L[/latex] is the limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] if, as the input values of [latex]x[/latex] approach [latex]a[/latex] (but do not equal [latex]a[/latex]), the corresponding output values of [latex]f\left(x\right)[/latex] get closer to [latex]L[/latex]. Note that the value of the limit is not affected by the output value of [latex]f\left(x\right)[/latex] at [latex]a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] must be real numbers. We write it as

Recall that [latex]y=3x+5[/latex] is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation [latex]\undersetx\to 2\mathrmlim\left(3x+5\right)=11[/latex], which means that as [latex]x[/latex] nears 2 (but is not exactly 2), the output of the function [latex]f\left(x\right)=3x+5[/latex] gets as close as we want to [latex]3\left(2\right)+5[/latex], or 11, which is the limit [latex]L[/latex], as we take values of [latex]x[/latex] sufficiently near 2 but not at [latex]x=2[/latex].

Figure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input [latex]x[/latex] within the interval [latex]6.9

We also see that we can get output values of [latex]f\left(x\right)[/latex] successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.

Figure 4 provides a visual representation of the left- and right-hand limits of the function. From the graph of [latex]f\left(x\right)[/latex], we observe the output can get infinitesimally close to [latex]L=8[/latex] as [latex]x[/latex] approaches 7 from the left and as [latex]x[/latex] approaches 7 from the right.

The values of [latex]f\left(x\right)[/latex] can get as close to the limit [latex]L[/latex] as we like by taking values of [latex]x[/latex] sufficiently close to [latex]a[/latex] but greater than [latex]a[/latex]. Both [latex]a[/latex] and [latex]L[/latex] are real numbers.

In the previous example, the left-hand limit and right-hand limit as [latex]x[/latex] approaches [latex]a[/latex] are equal. If the left- and right-hand limits are equal, we say that the function [latex]f\left(x\right)[/latex] has a two-sided limit as [latex]x[/latex] approaches [latex]a[/latex]. More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

In other words, the left-hand limit of a function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches [latex]a[/latex] is equal to the right-hand limit of the same function as [latex]x[/latex] approaches [latex]a[/latex]. If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

To visually determine if a limit exists as [latex]x[/latex] approaches [latex]a[/latex], we observe the graph of the function when [latex]x[/latex] is very near to [latex]x=a[/latex]. In Figure 5 we observe the behavior of the graph on both sides of [latex]a[/latex].

To determine if a right-hand limit exists, observe the branch of the graph to the right of [latex]x=a[/latex], but near [latex]x=a[/latex]. This is where [latex]x>a[/latex]. We see that the outputs are getting close to some real number [latex]L[/latex], so there is a right-hand limit.

Finally, we can look for an output value for the function [latex]f\left(x\right)[/latex] when the input value [latex]x[/latex] is equal to [latex]a[/latex]. The coordinate pair of the point would be [latex]\left(a,f\left(a\right)\right)[/latex]. If such a point exists, then [latex]f\left(a\right)[/latex] has a value. If the point does not exist, as in Figure 5, then we say that [latex]f\left(a\right)[/latex] does not exist.

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of [latex]x[/latex] approach [latex]a[/latex] from both sides. Then we determine if the output values get closer and closer to some real value, the limit [latex]L[/latex].

To create the table, we evaluate the function at values close to [latex]x=5[/latex]. We use some input values less than 5 and some values greater than 5 as in Figure 9. The table values show that when [latex]x>5[/latex] but nearing 5, the corresponding output gets close to 75. When [latex]x>5[/latex] but nearing 5, the corresponding output also gets close to 75.

We can estimate the value of a limit, if it exists, by evaluating the function at values near [latex]x=0[/latex]. We cannot find a function value for [latex]x=0[/latex] directly because the result would have a denominator equal to 0, and thus would be undefined.

We create Figure 10 by choosing several input values close to [latex]x=0[/latex], with half of them less than [latex]x=0[/latex] and half of them greater than [latex]x=0[/latex]. Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as [latex]x[/latex] approaches 0. If the function has a limit as [latex]x[/latex] approaches 0, state it. If not, discuss why there is no limit.

We can use a graphing utility to investigate the behavior of the graph close to [latex]x=0[/latex]. Centering around [latex]x=0[/latex], we choose two viewing windows such that the second one is zoomed in closer to [latex]x=0[/latex] than the first one. The result would resemble Figure 12 for [latex]\left[-2,2\right][/latex] by [latex]\left[-3,3\right][/latex].

Even closer to zero, we are even less able to distinguish any limits.The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function [latex]f\left(x\right)[/latex] as [latex]x[/latex] approaches 0.We write

Intuitively, we know what a limit is. A car can go only so fast and no faster. A trash can might hold 33 gallons and no more. It is natural for measured amounts to have limits. What, for instance, is the limit to the height of a woman? The tallest woman on record was Jinlian Zeng from China, who was 8 ft 1 in.1 Is this the limit of the height to which women can grow? Perhaps not, but there is likely a limit that we might describe in inches if we were able to determine what it was.

To put it mathematically, the function whose input is a woman and whose output is a measured height in inches has a limit. In this section, we will examine numerical and graphical approaches to identifying limits.

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