unit eq: what is the relationship between limits and the continuity of a function?

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