unit eq: what is the relationship between limits and the continuity of a function?
TRANSCRIPT
Limits and ContinuityUnit EQ:
What is the relationship between limits and the continuity of a function?
Limits Warm-Up/Activator
Which of the symbols ≥, >, ≤, and < go with [ or ]?Which of the symbols ≥, >, ≤, and < go with ( or )?
Limits – Touch LinkEssential Question:What is a limit and how are limits determined?
Definition of LimitIf f(x) becomes arbitrarily close to a single
number L as x approaches c from both sides, then
which is read: “________________________________”
What does this mean?
Lxfcx
)(lim
Example 1
Example 2a
Example 2b
Example 2c
1lim 2
1
x
x
Limits and Functions
1,0
1,lim
1 x
xxx1
1lim
1
x
xx
1
1lim
2
1
x
xx
Direct SubstitutionMany times the limit of f(x) as x approaches c
is simply f(c), this is called direct substitution.
Note: For direct substitution to work, the function must be continuous at c. We will further investigate the concepts of continuity and discontinuity in the next few lessons. Ex 2b and 2c were both discontinuous at c so direct substitution wouldn’t work.
Properties of Limits
Operations with LimitsLet b and c be real numbers and let n be a
positive integer. If the limits of f(x) and g(x) exist as x approaches c, then the following operations are valid.
1. Constant multiple:
2. Addition:
_______________)]([lim
xbfcx
_____________________)]()([lim
xgxfcx
Operations Continued3. Multiplication:
4. Division:
_____________________)]()([lim
xgxfcx
_______________)])(
)([lim
xg
xfcx
Operations Continued5. Power:
6. Radical:Note: In Property 6, if n is even and , then L must be positive.
n
cxxf )]([lim
____________)(lim
n
cxxf
Example 3
)32(lim 2
2xx
x
Example 3b
5
124lim
2
3
x
xx
Example 4Factor And Cancel MethodWhat is it?
1
1lim
3
1 x
xx
Example 5Factor & Cancel?
3
6lim
2
3 x
xxx
Example 6Factor & Cancel?
2
2lim
2
2 x
xxx
One-Sided LimitsExample
One-Sided LimitsA limit can fail (does not exist) when a
function approaches a different value from the left of c than it approaches from the right of c. This type of behavior can be described more concisely with the concept of a one-sided limit.
Left-Handed-
Right-Handed-
Find the limit as x approaches 0 from the right and left. (Example 7)
x
xxf
|2|)(
Existence of a Limit?If f is a function and c and L are real
numbers, then…
if and only if both the left and right limits are equal to L.
Lxfcx
)(lim
One-Sided Limit (Example 8)
Delivery Service Rates (Example 9)An overnight delivery service charges $8 for the
first pound and $2 for each additional pound. Let x represent the weight of a gift box and let f(x) represent the shipping cost.
Show that the limit of f(x) as x 2 does not exist.
Unbounded BehaviorA limit fails to exist when the left side limit
does not equal the right side limit. That also applies when f(x) increases or decreases without bound.
Example 10
2
3lim
2 xx
Last Limit Example