2-1: rates of change and limits Objectives: To evaluate limits numerically, graphically, and analytically. To use properties of limits

Discovering Limits Consider the function Find the domain of this function. Look at a table of values: As x gets closer and closer to 3, what does f(x) get close to? Is this the same as f(3)? As x gets closer and closer to 2 (be sure to look from left and right), what does f(x) get close to? Is this the same as f(2)? What’s going on?!?!?!?!?

Discovering Limits Even though f(x) is not defined at 2, the function’s value still appears to be approaching the same value from the left and the right. This value is called the limit of f(x) and is denoted by: Once you have this value, you have evaluated the limit. So,

Discovering Limits Given the function , complete the table: a f(a) 1 2

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.

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.

Evaluating numerically…using a table of values. Evaluate Try: x -.999 -.9999 -1 -1.0001 -1.001 f(x)

Use a graph and table of values to evaluate the limits below:

One-Sided Limits RIGHT-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)

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

Evaluate the following limits using the graph below:

Evaluate the following:

a. ) Graph the function b. ) Determine the LHL and the RHL c a.) Graph the function b.) Determine the LHL and the RHL c.) Does the limit exist? Explain.

Properties of Limits: If L, M, c and k are real numbers and and then: Sum and Difference rule: Product Rule: Constant Multiple Rule: Quotient Rule: Power Rule:

PRIZE ROUND Factor: x3+1 2. 8x3-27 3. t2+5t-6 4. 25x2 -64

Evaluating Algebraically Theorems: Polynomial and Rational Functions 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!!!

EXAMPLES

To evaluate limits algebraically: Try substitution. (c has to be in the domain). If you get 0/0, there is something you can do!! If substitution doesn’t work, factor if possible, simplify, then try to evaluate Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by conjugate. Simplify and evaluate. Use table or graph to reinforce your conclustion.

Examples: Evaluate the limit.

Evaluate the limit:

Evaluate.

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

Show

Useful Limits to Know!! Evaluate… 1. 2. 3.

Evaluate 1. 2. 3.