Unit 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 )?

Essential Question: What is a limit and how are limits determined?

Definition of Limit If 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?

Example 1

Example 2a

Example 2b

Example 2c

Limits and Functions

Direct Substitution Many 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 Limits Let 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:

Operations Continued 3. Multiplication: 4. Division:

Operations Continued 5. Power: 6. Radical: Note: In Property 6, if n is even and, then L must be positive.

Example 3

Example 3b

Example 4 Factor And Cancel Method What is it?

Example 5 Factor & Cancel?

Example 6 Factor & Cancel?

One-Sided Limits Example

One-Sided Limits A 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)

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.

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 Behavior A 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

Last Limit Example