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# 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 $\lim_{x\rightarrow a}f(x) = L_1$ and $\lim_{x\rightarrow a}g(x) = L_2$ .

Then:

1. $\lim_{x\rightarrow a} [f(x) + g(x)] = \lim_ {x\rightarrow a}f(x) + \lim_{x\rightarrow a}g(x) = L_1 + L_2$

• The limit of the sum is the sum of the limits.
• What is $\lim_{x \to 4} \sqrt{x} + x^4+3x$ ?

2. $\lim_{x\rightarrow a} [f(x) - g(x)] = \lim_ {x\rightarrow a}f(x) - \lim_{x\rightarrow a}g(x) = L_1 - L_2$

• The limit of the difference is the difference of the limits
• What is $\lim_{x \to 9} 2x^3-\sqrt{x}-\frac{x}{2}$ ?

3. $\lim_{x\rightarrow a} [f(x)g(x)] = ( \lim_ {x\rightarrow a}f(x)) ( \lim_{x\rightarrow a}g(x)) = L_1L_2$

• The limit of the product is the product of the limits.
• What is $\lim_{x \to \pi} \frac{x}{2}\cos(x)$ ?

4$\lim_{x\rightarrow a} \frac{f(x)}{g(x)} =\frac{\lim_{x\rightarrow a} f(x)} {\lim_{x\rightarrow a} g(x)} = \frac {L_1} {L_2} L_2 \neq 0$

• The limit of a quotient is the quotient of the limits (provided that the denominator does not equal zero.)
• What is $\lim_{x \to 5} \frac{3x+4}{2x^2-10}$ ?

5. If n is even:

•  The limit of the n th root is the n th root of the limit.
• What is $\lim_{x \to 4} \sqrt[3]{3x^2-10}$ ?

#### Results from the Previous Theorems

From the theorems listed above, we can conclude:

1. $\lim_{x\rightarrow a}k = k$

• if ) = , a constant function, then the values of ) do not change as is varied
• What is $\lim_{x \to 294} 17$ ?

2. $\lim_{x\rightarrow a} x= a$

• since ) = is an identity function (its input equals its output), then as → ) = → .
• What is $\lim_{x \to 113} x$ ?

3. $\lim_{x \rightarrow a} (k \cdot f(x)) = ( \lim_{x \rightarrow a} k) \cdot (\lim_{x \rightarrow a} f(x)) = k \cdot ( \lim_{x \rightarrow a} f(x))$

• What is $\lim_{x \to 3} 4(x^3-10)$ ?

4. $\lim_{x \rightarrow a} x^n = ( \lim_{x \rightarrow a} x)^n = a^n$

• What is $\lim_{x \to 2} (7x^2-23)^3$ ?

#### Theorem: The Limit of a Polynomial

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

$\lim_{x \rightarrow a}f(x) = c_n(a)^n + . . . + c_1(a) + c_0$
$\lim_{x \rightarrow a}f(x) = f(a)$

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: $\lim_{x \to 176} \sqrt[5]{x}$

2) Find the limit: $\lim_{x \to -3} \left( \frac{x^2 + 3x}{x + 3}\right)$

3) Given:

$\lim_{x \to c} f(x) = 7$
$\lim_{x \to c} g(x) = 3$

Find: $\lim_{x \to c} f(x) g(x)$

4) Given:

$f(x) = 4x^2 + 3x + 4$
$g(x) = 2x^2 -5x -2$

Find: $\lim_{x \to -2} f(g(x))$

5) Given:

$\lim_{x \to c} f(x) = -2$
$\lim_{x \to c} g(x) = 2$

Find: $\lim_{x \to c} \frac{f(x)}{g(x)}$

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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 $f(x) = \frac{p(x)} {q(x)}$ and any real number ,
$\lim_{x \rightarrow a} f(x) = \frac{p(a)} {q(a)} \text{ if } q(a) \neq 0$ .
However, if $\,\! q(a) = 0$ then the function may or may not have any outputs that exist.
• What is $\lim_{x \rightarrow 3} \frac{2 - x} {x - 2}$ ?

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:

1. $\lim_{x \rightarrow 4} \frac{x+6} {3-2x}$
2. $\lim_{x \rightarrow 5} \frac{3x-10} {x+5}$
3. $\lim_{x \rightarrow 1} \frac{4-3x} {x - 2}$

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

Try to find $\lim_{x \rightarrow 2} \frac{x^2 - 4} {x - 2}$ . What happens?

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

$\lim_{x \rightarrow 2} \frac{x^2 -4} {x - 2} = \lim_{x \rightarrow 2} \frac{(x - 2)(x + 2)} {x - 2} = \lim_{x \rightarrow 2} (x + 2) = 4$

What is $\lim_{x \rightarrow 3} \frac{2x - 6} {x^2 + x - 12}$ ?

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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  $\lim_{x \rightarrow 3} \frac{x + 1} {x - 3}$ ?

#### Practice

Find the following limits:

1. $\lim_{x \rightarrow 3} \frac{2x - 6} {x^2 + x - 12}$
2. $\lim_{x \to 1} \frac{-5x^2 +x +4}{x - 1}$
3. $\lim_{x \to -2} \frac{-x^2 + 2x + 8}{x + 2}$
4. $\lim_{x \to 1} \frac{-12x^2 + 12}{4x - 4}$
5. $\lim_{x \to \frac{-3}{2}} \frac{\frac{-4x - 3}{-2x+2} - \frac{5}{2}} {-2x -3}$
6. $\lim_{x \to \frac{1}{4}} \frac{-8x^2 - 2x + 1}{-4x + 1}$

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