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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

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 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.

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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). 

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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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