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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
To learn about the definition of limits, click here.
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 a 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:
 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 f ( x ) = k , a constant function, then the values of f ( x ) do not change as x is varied
 What is ?
2.
 since f ( x ) = x is an identity function (its input equals its output), then as x → a , f ( x ) = x → a .
 What is ?
3.
 What is ?
4.
 What is ?
Theorem: The Limit of a Polynomial
For any polynomial f ( x ) = c n x n + . . . + c 1 x + c 0 and any real number a ,
In other words, the limit of the polynomial is simply equal to f ( a ).
In your own words, describe how to find the limit of a poynomial.
.
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Practice
1) Find the limit:
2) Find the limit:
3) Given:
Find:
4) Given:
Find:
5) Given:
Find:
.
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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 a ,
 .
 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:
.
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