Limits

Investigate graphically the following limit:

lim x->2 ( x2 - 4 ) / ( x - 2 )

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

Turn on the CASIO CFX-9850G by pressing AC/ON.

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 ( X2 - 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 = ( x2 - 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 ( x2 - 4 ) = ( x + 2 )( x - 2 ) and we can simplify the expression to: ( x2 - 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.