numerical, graphical, and algebraic analysis of …...numerical, graphical, and algebraic analysis...

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 23

Mark Sparks 2012

Numerical, Graphical, and Algebraic Analysis of Functions

Given below are tables of values for different functions. Classify each function by type. Sketch a graph

of the function. Then, state as many specific properties, including the equation if possible, of each

function as you possibly can.

1.

2.

3.

x –5 –1 0 3 5 9

F(x) 31

37 –3 –5

319 –9

x –6 –4 –2 0 2 4

G(x) 5 1 –3 1 5 9

x –2 –1 0 1 2 3 4

H(x) –5 0 3 4 3 0 –5

spinners

Typewritten Text

spinners

Typewritten Text

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course 4

Mark Sparks 2012

4.

5.

x –4 –3 –2 1 6 13

J(x) Undefined –2 –1 0 1 2

x –6 –3 –1 0 2 4 6 10

K(x) 3.0156 3.125 3.5 4 7 19 67 1027


Mark Sparks 2012

6.

7.

x –11 –6 –1 0 2 4 6 10

M(x) 1.996 1.875 –2 –6 –30 –126 –510 –8190

x –1000 –3.001 –3 –2.999 0 0.999 1 1.001 1000

N(x) –1.997 –1.250 Undefined –1.249 1 2998 Undefined –3002 –2.003


Mark Sparks 2012

Understanding the Limit

A Numerical and Graphical Approach

The equation of the function graphed to the right

is9

352)(

2

2

x

xxxf . The coordinates of the hole in

the graph are 6

7,3 .

Pre-calculus Statements

Calculus Limit Notation

As x → −∞, the graph of f(x) → _________.

As x → ∞, the graph of f(x) → _________.

As x → –3 from the left, the graph of f(x) → _________.

As x → –3 from the right, the graph of f(x) → _________.

As x → 3 from the left, the graph of f(x) → _________.

As x → 3 from the right, the graph of f(x) → _________.

Based on what you have just seen, how might you informally define what the value of a limit represents

in terms of the graph?


Mark Sparks 2012

How is the numerical analysis above related in the graph of the function pictured below?


Mark Sparks 2012



Mark Sparks 2012

Limit Existence Theorem

Limits That Do Not Exist

Example #1

Find each of the following from the graph.

a) )(lim2

xfx

= b. )(lim2

xfx

=

c) f(2) =

d) Does )(lim2

xfx

exist or not? Why or why not?

Example #2


a) )(lim2

xfx

= b. )(lim2

xfx

=

c) f(2) =

d) Does )(lim2

xfx


Example #3


a) )(lim2

xfx

= b. )(lim2

xfx

=

c) f(2) =

d) Does )(lim2

xfx



Mark Sparks 2012

Based on what you have seen so far, does f(a) have to be defined in order for the )(lim xfax

to exist? Draw

and explain two different graphs to justify your reasoning. In both graphs, f(a) should be undefined but in

one graph, the limit should exist while in the second one, it should not exist.


Mark Sparks 2012


Mark Sparks 2012

Understanding the Limit

An Algebraic Approach

Consider the function, 442)(2

1 xxf , for a moment. The graph of f(x) is pictured below. From

the graph, determine the following limits.

When a function is defined and continuous at a value, x = a, how can )(lim xfax

be found analytically?

Find each of the following limits analytically.

a) 32lim 2

21

3

xx

x b.

32

25

3lim

x

x

x

c. 1

12

2

lim

x

x

x

d.

2sinlim2

e.

cos

32

lim

f. )11(loglim 89

xx

)(lim xfax

Find f(a) using the

equation. Find )(lim xf

axfrom

the graph.

)(lim0

xfx

)(lim2

xfx

)(lim10

xfx


Mark Sparks 2012

Analytically Finding Limits of Functions at Undefined Values

What happens if we try to evaluate the limits below by the direct substitution method that was used in the

previous six examples?

32

34lim

2

2

3

xx

xx

x

32

34lim

2

2

1

xx

xx

x

32

34lim

2

2

1

xx

xx

x

Just because a function is undefined at a value of x does not mean

that a conclusion cannot be reached about the limit. Consider the

rational function above. From the graph of the function pictured

to the right, what is the value of each limit below?

_______32

34lim

2

2

3

xx

xx

x

_______32

34lim

2

2

1

xx

xx

x

_______32

34lim

2

2

1

xx

xx

x

The task now is to determine how to find these limits analytically. How was it that we found the

discontinuities of a rational function in pre-calculus?

We will perform the same algebraic analysis to find the limit of the removable, point discontinuities.

Let’s do this Cancellation Process below.

32

34lim

2

2

3

xx

xx

x


Mark Sparks 2012

Based on our knowledge from pre-calculus, we know that if a rational function has a non-removable

infinite discontinuity, graphically a ____________________________ exists. Since the y – values do not

approach one specific value from both sides at a __________________________, then the limit does not

exist. However, we can determine if the one sided limits approach −∞ or ∞. In order to do this

analytically, we will marry the numerical, graphical, and algebraic approaches. For each limit below,

determine the sign of the simplified function at the value to the right or the left of x = 1.

32

34lim

2

2

1

xx

xx

x

32

34lim

2

2

1

xx

xx

x

Value of

x to the left

of x = 1

Simplified function

1

1

x

x

0.9

The graph of a function 3

21)(

x

xxg is pictured to the right. Often, rationalization can be used

to evaluate a limit analytically. Find the following limit.

3

21lim

3

x

x

x

Value of

x to the right

of x = 1

Simplified function

1

1

x

x

1.1


Mark Sparks 2012

Write the equation of the piece-wise defined function pictured to the right.

Constraint of

Equation of Each Piece Each Piece

)(xf


Mark Sparks 2012

Consider the function below to find each limit. If a limit does not exist, state why.

2,1

2,32)(

21

2

xx

xxxxG

a) )(lim2

xGx

b) )(lim2

xGx

c) )(lim2

xGx

Find each of the following limits analytically. Show your algebraic analysis.

a. x

x

ex 2

lnlim

b. xxx

2lim 2

52

5

c.

cos2sinlim 2

d. 2

tanlim

35

e. 42

6lim

2

2

x

xx

x


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f. 9

5lim

23

x

x

x

g. 32

278lim

3

23

x

x

x

h. 2

152lim

2

x

x

x

i. 1

121lim

2

1

x

x

x


Mark Sparks 2012

j. x

xx

x

12

1

0lim

k. 4

273lim

2

2

2

x

xx

x

l. 3

52lim

3

x

x

x

m. 3

52lim

3

x

x

x

If f(x) = 2x2 – 3x + 4, find

h

xfhxf

h

)()(

0lim

.


Mark Sparks 2012

Properties of Limits

Suppose Lxfax

)(lim and Mxgax

)(lim . Find each of the following limits in terms of L

and M.

1. )()(lim xgxfax

2. )()(lim xgxfax

3. )(

)(lim

xg

xf

ax

4. )()(lim xgxfax

5. )(lim xfcax

6. p

axxf )(lim

7. cax

lim


Mark Sparks 2012

Graph of f(x) Graph of g(x)

Find each of the following limits applying the properties of limits. If a limit does not exist,

state why.

)()(lim2

xgxfx

)(3)(2lim1

xgxfx

)()(lim3

xgxfx

)(

)(2lim

6 xg

xf

x

)()(2lim4

xgxfx

22

)(lim xfx

)(2lim2

xgx

)(

)(lim

2 xg

xf

x

)()(lim1

xgxfx

numerical, graphical, and algebraic analysis of …...numerical, graphical, and algebraic analysis...

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