Exploring Limits

In this exploration, we will examine limits from a numerical and visual point of view.

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Finding Limits Numerically

Let's investigate the behavior of f(x) = x^2 – 1 near x = 2.

Select Calculators - Expression Calculator from the Work menu.
Enter f(2.1). Press the Enter key or click the Evaluate button in the lower left corner of the Expression Calculator window. The value of the function f(x) at x = 2.1 will be written to the next line in the Expression Calculator window.
Change f(2.1) to f(2.01). Press the Enter key or click the Evaluate button. The value of the function f(x) at x = 2.01 will be written in the Expression Calculator window.
Change f(2.01) to f(2.001). Press the Enter key or click the Evaluate button.
Continuing in this manner, find the values of f(2.0001) and f(2.00001).

As the value of x gets close to 2 from the right, what value does f(x) approach?

Find the values of f(1.9), f(1.99), f(1.999), f(1.9999), and f(1.99999).

As the value of x gets close to 2 from the left, what value does f(x) approach?

We can now guess that

lim f(x) = 3
x->2

In fact, we also note that

lim f(x) = 3 = f(2),
x->2

that is, the limit of f(x) as x->2 is equal to f(2). In this case, we can say that f(x) is continuous at x = 2.

As a second example, let's examine the behavior of g(x) near x = 2. Note that g(x) is not defined at x = 2. Why is g(x) is not defined at x = 2?

Enter g(2.1) after the last in the Expression Calculator window. Press the Enter key.
Find the value of g(2.01), g(2.001), g(2.0001), g(1.9), g(1.99), g(1.999), and g(1.9999).

Limit of g(x)

As the value of x gets close to 2, what value does g(x) approach?

We can now guess that

lim g(x) = 4
x->2

Note that for this example, g(x) is not defined at x = 2 but the limit still exists. In this case, we can say that g(x) is not continuous at x = 2.

Exploring Limits Graphically

Plot f(x) on -3x3 and -8y8.
Click the Rectangular Tracker (Trace) tool's icon .
Position the arrow cursor over the up arrow part of the tracker, press and hold down the mouse button, and drag the tracker to the right.
As you drag the tracker near x = 2, notice the displayed values of x and f(x). Also, observe that the values of f(x) get close to 3.

Graphical Limit of f(x)

Plot g(x) on -3x3 and -8y8.
If the Rectangular Tracker is not displayed, click the Rectangular Tracker (Trace) tool's icon.
Click in the Domain & Range window to make it the active window.
Enter 1.9 into the "x =" cell.
Click the Enter button or press the Enter key.
Note that the tracker moves to the point on the curve at x = 1.9 and the value of y = g(1.9) is displayed in the "y =" cell in the Domain & Range window and also in the Graph window.
Enter 1.99 into the "x =" cell. Click the Enter button or press the Enter key and note the value of y = g(1.99). Repeat this for x = 1.999 and x = 1.9999.
Enter 2 into the "x =" cell. Click the Enter button or press the Enter key and note that g(2) is not defined.

A Removable Discontinuity

Plot r(x) on -2x2 and -4y4. Note the change in the plotting domain and range.
If the Rectangular Tracker is not displayed, click the Rectangular Tracker (Trace) tool's icon.
Drag the Rectangular Tracker slowly to the right past the value x = -1.
Note that at x = -1 the graph jumps to the value 1.

If the tracker does not jump at x = -1, you will need to change the resolution of the Graph window by doing the following: Select Axes Style... from the Graph menu, set the Horizontal Axis Pixel Length to 376, set the Vertical Axis Pixel Length to 376, click the OK button, plot r(x), and try dragging the tracker across x = -1 again.

What is

lim r(x) = ?
x->-1

How could you change r(x) to make it continuous at x = -1?

One-sided Limits

Plot s(x) on -2x2 and -4y4.
If the Rectangular Tracker is not displayed, click the Rectangular Tracker (Trace) tool's icon.
Use the Rectangular Tracker to investigate the limit of s(x) as x gets close to 1 from the left by dragging the tracker near x = 1 but for values of x less than 1.

Left limit of s(x)

What is

lim s(x) = ?
x->1-

Use the Rectangular Tracker to investigate the limit of s(x) as x gets close to 1 from the right by dragging the tracker near x = 1 but for values of x greater than 1.

Right limit of s(x)

What is

lim s(x) = ?
x->1+

Does

lim s(x)
x->1

exist?

Infinite Limits

Plot u(x) on 0x2 and 0y100. Note the change in the plotting domain and range.
If the Rectangular Tracker is not displayed, click the Rectangular Tracker (Trace) tool's icon.
Use the Rectangular Tracker to investigate the limit of u(x) as x gets close to 1.
Note that as the tracker gets close to x = 1, you can read the values of u(x) in the "y =" cell in the Domain & Range window.

Limit of u(x) Domain and Range Window

What is

lim u(x) = ?
x->1

Plot a(x) on 0 x 2 and -100 y 100. If a(x) is not active, that is, it doesn't have a checkmark to the left of R, then click in the space to the left of R to display a checkmark and make a(x) active.
If the Rectangular Tracker is not displayed, click the Rectangular Tracker (Trace) tool's icon.
Use the Rectangular Tracker to investigate the limit of a(x) as x gets close to 1. Note that as the tracker gets close to x = 1, you can read the values of a(x) in the "y =" cell in the Domain & Range window.

Left limit of a(x)

Right limit of a(x)

What is

lim a(x) = ?
x->1-

What is

lim a(x) = ?
x->1+

What is

lim a(x)?
x->1

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