2-1: RATES OF CHANGE AND LIMITS

Objectives:• To evaluate limits numerically,

graphically, and analytically.• To use properties of limits

Average Speed or Rate of Change

Distance Covered

Elapsed time

A rock breaks free from the top of a cliff. What is the average speed during the first 2 seconds?? (y=16t2)

t

y

What if I wanted to know the speed at EXACTLY 2 seconds (use same function)?

Let us use t = 2 and t= 2 + h.

Definition of a Limit

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement

Means “The values f(x) of the function f approach or equal L as the values of x approach (but do not necessarily equal) c.

Lxfcx

)(lim

In other words…If the values of f(x) approach the number L as x approaches a from both the left and the right, we say that the limit L as x approaches a exists and

**Please note..a limit describes how the outputs of a function behave as the inputs approach some particular value. It is NOT necessarily the value of the function at that x value.

Lxfcx

)(lim

Evaluating numerically…using a table of values.

Evaluate

Try:

x -.999 -.9999 -1 -1.0001 -1.001

f(x)

1

1lim

2

1

x

xx

2

23lim

2

2

x

xxx

One-Sided LimitsRIGHT-HAND LIMIT (RHL)

(The limit of f as x approaches c from the right)

LEFT-HAND LIMIT(LHL)

(The limit of f as x approaches c from the left)

)(lim xfcx

)(lim xfcx

IN ORDER FOR A LIMIT TO EXIST, THE FUNCTION HAS TO BE APPROACHING THE SAME VALUE FROM BOTH THE LEFT AND THE RIGHT (LHL and RHL must exist and be equal)

IF =

THEN

)(lim xfcx

)(lim xfcx

Lxfcx

)(lim

a.) Graph the functionb.) Determine the LHL and the RHLc.) Does the limit exist? Explain.

2,2/

2,2

2,3

)(,2

xx

x

xx

xfc

Properties of Limits: If L, M, c and k are real numbers and

and then:

1. Sum and Difference rule:

2. Product Rule:

3. Constant Multiple Rule:

4. Quotient Rule:

5. Power Rule:

Lxfcx

)(lim Mxgcx

)(lim

0,)(lim

0,)(

)(lim

)(lim

))()((lim

))()((lim

sLxf

MM

L

xg

xf

kLxkf

MLxgxf

MLxgxf

s

r

s

r

cx

cx

cx

cx

cx

Evaluating Algebraically

Theorems: Polynomial and Rational Functions

1. If f(x) = anxn + an-1xn-1+…+a0 is any polynomial function and c is a real number, then

SUBSTITUTE!!!!!!

2.If f(x) and g(x) are polynomials and c is a real number, then

SUBSTITUTE!!!

01

1 ...)()(lim acacacfxf nn

nn

cx

0)(,)(

)(

)(

)(lim

cgcg

cf

xg

xfcx

EXAMPLES

4

3lim.2

)12(lim.1

2

1

3

2

x

x

xx

x

x

PRIZE ROUND

Factor:

1. x3+1

2. 8x3-27

3. t2+5t-6

4. 25x2 -64

To evaluate limits algebraically:

1. Try substitution. (c has to be in the domain). If you get 0/0, there is something you can do!!

2. If substitution doesn’t work, factor if possible, simplify, then try to evaluate

3. Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by conjugate. Simplify and evaluate.

4. Use table or graph to reinforce your conclustion.

Examples: Evaluate the limit.

h

h

x

x

xx

xx

xx

h

x

x

x

11

0

4

2

2

5

33lim.4

4

2lim.3

coslim.2

158

103lim.1

Evaluate the limit:

n

n

p

p

x

xx

n

p

x

93lim.3

1

1lim.2

9

43lim.1

2

0

3

1

2

2

2

Evaluate.

2,23

2,2lim.3

2

1lim.2

2

13lim.1

2

2

3

2

2

1

xx

xx

x

x

x

x

x

x

x

Sandwich (Squeeze) Theorem

If g(x) < f(x) < h(x) when x is near c (except possibly at c) and

THEN

Lxhxgcxcx

)(lim)(lim

Lxfcx

)(lim

Show 020coslim 2

0

xx

x

Useful Limits to Know!! Evaluate…

1.

2.

3.

c

x

x

x

x

cx

x

x

lim

cos1lim

sinlim

0

0

Evaluate

1.

2.

3.

x

x

x

x

x

x

x

tanlim

2sinlim

15lim

0

0

15