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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