INCREASING AND DECREASING FUNCTIONS



SECTION 1.2:

FINDING LIMITS GRAPHICALLY AND NUMERICALLY

Goals: The Student Will Be Able To:

❖ Create a useful table of values, and use it to evaluate a limit (numerical analysis)

❖ Create a hand-sketched graph, and use it to evaluate a limit (graphical analysis)

❖ Use a graphing utility to set up a table and trace graphs, so to evaluate a limit

❖ Determine whether the graphical or numerical approach is more appropriate

❖ Determine and explain when and how certain types of limits do not exist (jump, infinite, oscillating)

❖ Acquire and use new associated terminology and symbolic notation

INTUITIVE SENSE OF A LIMIT

(2.5, 4)

(2.9, 4.8)

(2.99, 4.98)

(2.9999, 4.9998)

(3.0001, 5.0002)

(3.01, 5.02)

(3.1, 5.2)

(3.5, 6)

GENERAL DESCRIPTION OF A LIMIT

The word “LIMIT” is abbreviated: ”lim”

It is used in reference to the value of a function or expression. It denotes the “y” value that a function or expression is getting closer and closer to, as corresponding “x” values approach a fixed number.

Symbolically:

The “c” is some particular/specific “x” value. One must consider values a little bit bigger and a little bit smaller than this “c” value in evaluating a limit. A limit disregards what is happening at “c”.

The “L” is the resulting “y” value, known as the limit. It is what appears to be the value that f(x) is “settling in on”, as you evaluate it in a close “neighborhood” of values around “c”.

EXAMPLES OF LIMIT EVALUATIONS

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

NUMERICAL EVALUATION OF A LIMIT

Consider the following function: [pic]

Can you evaluate the function for various “x” values? …

f(2); f(3); f(0); f(-1); f(-2) … f(1) ???

[pic] [pic]

[pic] [pic] [pic]

Is there one value of “x” which will cause a problem? …

But of course there is!!! There is one value not in the DOMAIN of f(x) ( [pic]

[pic]

There is NOT a corresponding “y” value for “x”, when x = 1. But we can examine the “y” values, for “x” values that are “near” (in a neighborhood) around x = 1. This is the idea of a LIMIT.

[pic]

Instead of asking: [pic]… Ask: [pic]

One way to answer this is by using a numerical approach (there will be others).

|X |.9 |.999 |1 |1.001 |1.1 |

|f(x) |2.710 |2.997 |??? |3.003 |3.310 |

You can move arbitrarily closer and closer to x = 1, and witness how y will move ever closer to y = 3.

Thus we can say: [pic]

“The limit of the function as x approaches 1 is 3”

Here … try one on your own… [pic]

Do it numerically!!!

GRAPHICAL EVALUATION OF A LIMIT

Consider the same function again:[pic]

However, this time, examine the LIMIT question by looking at the graph. Let’s use our TI83 to do this.

Enter into Y= [pic] then

ZOOM…4:ZDecimal…

WINDOW…Xmin/2…

Xmax/2…Ymin=0…

Ymax*2…GRAPH

OH MY!!! Is your calculator broken? No, the “hole” in the graph is the visual effect of the break in the domain at x = 1.

Use TRACE to explore around the “hole”. See what happens to “y”, when the cursor is at x = 1.

Use TRACE…ZOOM…2:Zoom In…ENTER a few more times to visually inspect the “hole” neighborhood even more closely. Does the graph confirm that the limit is 3?

EXAMPLE 1:

Find the limit numerically, and then graphically:

[pic] Do you see why f(0) is a problem?

|X |-0.01 |-0.0001 |0 |0.0001 |0.01 |

|f(x) |1.9950 |1.9999 |??? |2.0001 |2.0050 |

It should be quite obvious that as x ( 0; y or f(x) (2

[pic] … Confirm this by graphing.

ZOOM…4:ZDecimal…

WINDOW…Ymin=0…

GRAPH

The “hole” in the graph will only be visible if you turn off your axes. It should become clear that the “hole” is at (0,2).

EXAMPLE 2:

Find the limit of the “piecewise function”:

[pic]

[pic]

We can look at it both graphically and numerically. The graphical evaluation makes better sense. What is the “y” value as x approaches 2 from the left and the right? The “y” value stays at a constant of 1, regardless of how close you are to x = 1. The big point is that, what is happening AT x = 1 is irrelevant!!! Imagine 2 bugs walking on the curve, approaching the “hole” from the left and right. They will be able to shake hands. As they “meet over the hole”, what are their “y” values?

[pic] even though [pic]

Numerically:

|X |1.9 |1.9999 |2 |2.0001 |2.1 |

|f(x) |1 |1 |0 |1 |1 |

EXAMPLE 3:

Show that the following limit does not exist.

Consider the graph, and what is happening for positive x values that are getting closer to 0.

Now consider the graph for the negative x values that are getting closer to 0.

Can you imagine how frustrated the two bugs are, who cannot shake hands?

Since the “right-hand” y value is at 1, and the “left-hand” y value is –1, the overall limit DOES NOT EXIST! … because the left and right behavior differs.

Numerically:

|X |-0.01 |-0.0001 |0 |.0001 |0.01 |

|f(x) |-1 |-1 |??? |1 |1 |

EXAMPLE 4:

Discuss the existence of the limit:

Consider the graph, and what is happening for positive x values that are getting closer to 0. “y” is getting bigger and bigger, with no end in sight.

Now consider the graph for the negative x values that are getting closer to 0. The same thing is happening to “y”

The two bugs are frustrated again, but for a different reason. They never settle in on a final “y” value”. And, it’s as if they have a thin glass wall between their feet.

Numerically:

|X |-0.01 |-0.0001 |0 |.0001 |0.01 |

|f(x) |10000 |100000000 |??? |100000000 |10000 |

…or another way to better say this is:

… the function “increases without bound”.

EXAMPLE 5:

Discuss the existence of the limit:

[pic]

Consider the graph, and what is happening for positive or negative x values that are getting closer to 0. “y” appears to be oscillating back and forth between extremes of 1 and –1.

Again, since there is not one specific value that the function is settling in to, there is no limit.

The numerical approach can confirm this, as well. Here is a unique numerical view from the right to illustrate the unending oscillation.

Numerically:

|x |[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |x ( 0 |

|[pic] |1 |-1 |1 |-1 |1 |-1 |Limit d.n.e. |

-----------------------

(_______,_______)

[pic]

From the left

From the right

1

3

1

1

2

[pic]

1

-1

Disagreement

[pic]

[pic]

[pic]

1

-1

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download