Objectives: To evaluate limits numerically, graphically, and analytically. To evaluate infinite limits

x f(x) * USING YOUR CALCULATORS, MAKE A TABLE OF VALUES TO FIND THE VALUE THAT f(x) IS APPROACHING AS x IS APPROACHING 1 FROM THE LEFT AND FROM THE RIGHT. f(x) = 3x + 1 “As x is approaching 1 from the left” “As x is approaching 1 from the right”

X

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 **You can use a table of values to find a limit by taking values of x very, very, very close to a on BOTH sides and see if they approach the same value

* 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 (but it could be). WHAT???????????????? Yes, this is true

RIGHT-HAND LIMIT (RHL) (The limit as x approaches a from the right) LEFT-HAND LIMIT(LHL) (The limit as x approaches a 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 = RHL) =

* Graph the following function. Then find the limit.

Look at a table of values and the graph of What happens as x approaches 2? DOES NOT EXIST

is not a number. It is used to describe a situation where something increases or decreases without bound (gets more and more negative or more and more positive)

A LIMIT DOES NOT EXIST (DNE) WHEN: 1. The RHL and LHL as x approaches some value a are BOTH or BOTH -. We write or, but the limit DNE. 2. The RHL as x approaches some value a is and the LHL as x approaches the same value is - or vice versa. 3. LHL ≠ RHL (The fancy dancy explanations are on page 154)

Find all the zeros: 2x 3 +x 2 -x

1. Sum/Difference Rule: 2. Product Rule: 3. Constant Multiple Rule: 4. Quotient Rule: 5. Power Rule:

* If one of the limits for one of the functions DNE when using the properties, then the limit for the combined function DNE.

1. If p(x) is a polynomial, then 2., where c is a constant 3.

* Take a look at p. 165 # 25 and 30.

1. Try substitution (If a is in the domain of the function this works). If you get 0/0 when you substitute, there is something you can do to simplify!! 2. If substitution doesn’t work, simplify, if possible. Then evaluate limit. 3. Conjugate Multiplication: If function contains a square root and no other method works, multiply numerator and denominator by the conjugate. Simplify and evaluate. You can always use a table or a graph to reinforce your conclusion

Factor the following: 1. x x x 2 -9

Lets do some examples together, shall we???? Handout—Finding Limits Algebraically—Classwork I do #1,3,5,8,10,11 with you You try #2,4,7,9

Evaluate the limit:

* Evaluate the limit:

* What is the function’s value approaching as the x values get larger and larger in the positive direction? Larger and larger in the negative direction?

FOR ANY POSITIVE REAL NUMBER n AND ANY REAL NUMBER c : and TO FIND THE FOR ANY RATIONAL FUNCTION, DIVIDE NUMERATOR AND DENOMINATOR BY THE VARIABLE EXPRESSION WITH THE LARGEST POWER IN DENOMINATOR.

WHEN WE ARE EVALUATING THESE LIMITS AS x ±∞, WHAT ESSENTIALLY ARE WE FINDING? * WE LEARNED IT IN PRE-CALC WHEN WE GRAPHED RATIONAL FUNCTIONS * WHAT DOES THE END-BEHAVIOR OF A FUNCTION TELL US? * IT BEGINS WITH AN “H”

* DEFINTION OF HORIZONTAL ASYMPTOTE * THE LINE y=b IS A HORIZONTAL ASYMPTOTE OF THE GRAPH OF y=f(x) IF EITHER OR

***NOTE: A function can have more than one horizontal asymptote. Take a look at these graphs.

1. If the degree of the numerator is less than the degree of the denominator, the limit of the rational function is If the degree of the numerator is = to the degree of the denominator, the limit of the rational function is the ratio of the leading coefficients. 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function approaches ±∞.

* LOOK AT THE GRAPH OF

The line x=a is a vertical asymptote of y=f(x) if either: OR

1. Sum/difference: 2. Product: 3. Quotient:

3. Evaluate the limit: