Limits

Investigate graphically the following limit:

lim x->2 ( x^{2} - 4 ) / ( x - 2 )

Let’s graph the function using the initial viewing window.

Turn on the **CASIO CFX-9850G** by pressing

Use the cursor arrows to highlight the GRAPH menu and press **EXE **or just press **5**
when at the main menu screen.

In the Y1 slot enter ( X^{2} - 4 ) ¸ ( X - 2 ) and
press **EXE** to store it in memory.

Press **SHIFT** and **F3** (V-Window) to enter the viewing window screen, and
press **F1** (INIT) to select the initial viewing window parameters.

Press **EXIT** to return to the function list. Make sure that no other Y-slot is
highlighted, i.e. selected for drawing.

Press **F6** (DRAW) to sketch the graph.

As we can see this viewing window is not the best to study the given limit because we cannot see what happens to y as x approaches 2. We can scroll to the right and up, using the cursor arrows (right cursor arrow and up cursor arrow), once to the right and twice up to get a much better picture.

Alternatively, we can let the calculator set the viewing window automatically for the
given x ranges by pressing **F2** (Zoom) **F5** (AUTO).

Either way we see a hole in the graph for x=2.

For future use, lets store the function and the viewing window into the calculator
memory. Press **SHIFT** **F3** (V-Window), **F4** (STO), **F1** (V-W1), and **EXIT**.
From the function list press **F5** (GMEM), **F1** (STO), **F1** (GM1) and **EXIT**.

Press **F6** (DRAW) to get back to the graph. Tracing the function and zooming in we
can make an educated guess for this limit and say that as x approaches 2, the function y =
( x^{2} - 4 ) / ( x - 2 ) approaches 4. For tracing press **F1** (Trace) and
the right left cursor keys. For zooming in press **F2** (Zoom), **F3** (IN).

This limit is not surprising for the astute student since ( x^{2} - 4 ) = ( x +
2 )( x - 2 ) and we can simplify the expression to: ( x^{2} - 4 ) / ( x - 2 ) = (
x + 2 ). However we need to add that x ? 2 in this new expression. Failing to do this
would be a mistake.

To illustrate this to your students simply recall the previously saved viewing window
by pressing **SHIFT** **F3** (V-Window) **F5** (RCL) **F1** (V-W1) and **EXIT**.
Enter X + 2 into Y2 and press **EXE** and **F6** (DRAW). Y2 fills the gap left open
by Y1.

Press **EXIT** to get back to the function list.

Investigate graphically the following limit:

lim x->0 sin x / x

Before going any further clear all functions in the function list. Use the up and down
cursor keys to scroll to Y1 and press **F2** (DEL) **F1** (YES). Scroll to Y2 and
press **F2** (DEL) **F1** (YES).

Make sure that the calculator is on Radian mode by pressing **SHIFT** **SETUP**
(MENU); scroll down to angle and press **F2** (Rad). Press **EXIT**.

Enter sin x ¸ x into Y1 and press **EXE** to store it in
memory.

We will graph sin x ¸ x using the trigonometric window, so
press **SHIFT** and **F3** (V-Window) to enter the viewing window screen and then
press **F2** (TRIG) to select the initial viewing window.

Press **EXIT** to return to the function list. Press **F6** (DRAW) to sketch the
graph of Y1.

This graph should also have a hole at x=0, however the y axes is on the way. To get rid
of the axis press **SHIFT** **SETUP** (MENU) and scroll down to axes. Press **F2**
(Off) to turn the axes off. Press **EXIT** and **F6** (DRAW). We see the hole as we
expected. Tracing and Zooming in we will be able to make an educated guess that this limit
is 1.

Use the Squeeze Theorem graphically to illustrate that the limit is in fact 1.

Make sure you have the trigonometric window by pressing **SHIFT** **F3**
(V-window) **F2** (TRIG).

Press **EXIT**.

Enter 1 into Y2. Press **EXE**. Color Y2 orange by pressing **F4** (COLR) **F2**
(Orng).

Enter cos X into Y3. Press **EXE**. Color Y3 orange by pressing **F4** (COLR) **F2**
(Orng).

Press **EXE**.

You will see a colorful illustration of the squeeze theorem that can be used in class to explain its usefulness.

Press **EXIT**.

Press **MENU**.

Turn off the **CASIO CFX-9850G** by pressing