# Polynomial Function Limits

## Limits involving function operations, constants, and values of polynomials.

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Practice Polynomial Function Limits

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Finding Limits

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##### Guiding Questions

How do we find limits of polynomials and rational functions?

What are some important things to remember about finding limits?

### Important Theorems of Limits

#### Vocabulary

Limit theorems: a series of statements describing the effects of various mathematical operations on limits

The limit theorems are listed below.

#### Theorems of Limits

Let be a real number and suppose that  and  .

Then:

1.

• The limit of the sum is the sum of the limits.
• What is  ?

2.

• The limit of the difference is the difference of the limits
• What is  ?

3.

• The limit of the product is the product of the limits.
• What is  ?

4

• The limit of a quotient is the quotient of the limits (provided that the denominator does not equal zero.)
• What is  ?

5. If n is even:

[Figure1]
•  The limit of the n th root is the n th root of the limit.
• What is  ?

#### Results from the Previous Theorems

From the theorems listed above, we can conclude:

1.

• if ) = , a constant function, then the values of ) do not change as is varied
• What is  ?

2.

• since ) = is an identity function (its input equals its output), then as → ) = → .
• What is  ?

3.

• What is  ?

4.

• What is  ?

#### Theorem: The Limit of a Polynomial

For any polynomial ) = c + . . . + c x + c and any real number ,

In other words, the limit of the polynomial is simply equal to ).

In your own words, describe how to find the limit of a poynomial.

.

#### Practice

1) Find the limit:

2) Find the limit:

3) Given:

Find:

4) Given:

Find:

5) Given:

Find:

.

### Rational Function Limits

#### Vocabulary

Rational function: any function which can be written as the ratio of two polynomial functions.

Discontinuous function: a function that exhibits breaks or holes when graphed.

#### Theorem: The Limit of a Rational Function

For the rational function  and any real number ,
.
However, if  then the function may or may not have any outputs that exist.
• What is  ?

In your own words, describe how to find the limit of a rational function (assume the denominator does not equal 0).

.

Find the following limits:

#### What happens when the denominator equals zero?

Try to find  . What happens?

When the denominator equals 0 when the limit statement is plugged in, we have to factor.

What is  ?

.

Sometimes factoring isn't possible. In that case, check the limit from both sides. If they are not equal, the limit does not exist.

What is   ?

#### Practice

Find the following limits:

.