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Polynomial Function Limits

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

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Polynomial Function Limits

You may recall that many of the rules you have learned to use when solving basic algebra equations often apply to the manipulation of complex equations also. You will find in this lesson that a similar concept applies to operations on limits.

Polynomial Function Limits

In this lesson, you will evaluate the limits of functions from an algebraic perspective. First you will require the proper tools for the job, the theorems necessary to calculate limits. These theorems are outlined in the two boxes below.

Box #1: Important Theorems of Limits

Let a be a real number and suppose that \begin{align*}\lim_{x\rightarrow a}f(x) = L_1 \end{align*} and \begin{align*}\lim_{x\rightarrow a}g(x) = L_2 \end{align*}.


1. \begin{align*} \lim_{x\rightarrow a} [f(x) + g(x)] = \lim_ {x\rightarrow a}f(x) + \lim_{x\rightarrow a}g(x) = L_1 + L_2\end{align*}, meaning the limit of the sum is the sum of the limits.

2. \begin{align*} \lim_{x\rightarrow a} [f(x) - g(x)] = \lim_ {x\rightarrow a}f(x) - \lim_{x\rightarrow a}g(x) = L_1 - L_2\end{align*}, meaning the limit of the difference is the difference of the limits.

3. \begin{align*} \lim_{x\rightarrow a} [f(x)g(x)] = ( \lim_ {x\rightarrow a}f(x)) ( \lim_{x\rightarrow a}g(x)) = L_1L_2\end{align*}, meaning the limit of the product is the product of the limits.

4. \begin{align*} \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\end{align*}, meaning the limit of a quotient is the quotient of the limits (provided that the denominator does not equal zero.)

5. \begin{align*} \lim_{x\rightarrow a} \sqrt[n]{f(x)}= \sqrt[n] { \lim_{x\rightarrow a} f(x)} = \sqrt[n] L_1 L_1 > 0\end{align*} if n is even, meaning the limit of the nth root is the nth root of the limit.

Other useful results follow from the above theorems:

Box #2

1. If a and k are real numbers, then \begin{align*}\lim_{x\rightarrow a}k = k\end{align*}. That is, if f(x) = k, a constant function, then the values of f(x) do not change as x is varied, thus the limit of f(x) is k.

2. If a is a real number then \begin{align*}\lim_{x\rightarrow a} x= a \end{align*}. That is, since f(x) = x is an identity function (its input equals its output), then as xa, f(x) = xa.

3. \begin{align*}\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))\end{align*}

4. \begin{align*}\lim_{x \rightarrow a} x^n = ( \lim_{x \rightarrow a} x)^n = a^n\end{align*}

Finally, we have one more theorem:

Theorem: The limit of a polynomial

For any polynomial f(x) = cnxn + . . . + c1x + c0 and any real number a,
\begin{align*}\lim_{x \rightarrow a}f(x) = c_n(a)^n + . . . + c_1(a) + c_0\end{align*}
\begin{align*}\lim_{x \rightarrow a}f(x) = f(a)\end{align*}
In other words, the limit of the polynomial is simply equal to f(a).


Example 1

Use the theorems above to find \begin{align*}\lim_{x \rightarrow 1}(x^2 - 3x + 4)\end{align*} and justify each step.

You can verify your practice application of the theorems described above by simply substituting x = 1 directly into the polynomial.

Using Equation (1) of Box #1 (the limit of the sum is the sum of the limits) we get

\begin{align*}\lim_{x \rightarrow 1}(x^2 - 3x + 4) = \lim_{x \rightarrow 1}x^2 + \lim_{x \rightarrow 1}(-3x) + \lim_{x \rightarrow 1} 4\end{align*}

From Equation (4) of Box #2, the first term becomes

\begin{align*}\lim_{x \rightarrow 1}x^2 = (1)^2 = 1\end{align*}

From Equations (2) and (3) of Box #2, the second term becomes

\begin{align*}\lim_{x \rightarrow 1}(-3x) = -3 \lim_{x \rightarrow 1}x = (-3)(1) = -3\end{align*}

Finally, from Equation (1) of Box #2, the third term becomes

\begin{align*}\lim_{x \rightarrow 1}4 = 4\end{align*}

Thus the limit of the above polynomial is

\begin{align*}\lim_{x \rightarrow 1}(x^2 - 3x + 4) = 1 + (-3) + 4 = 2\end{align*}

Example 2

Find \begin{align*}\lim_{x\rightarrow 3} (4x^3 - 4x - 5)\end{align*}.

According to the theorem above, \begin{align*}\lim_{x\rightarrow 3}(4x^3 - 4x - 5) = 4(3)^3 - 4(3) - 5 =91\end{align*}

Example 3

Find \begin{align*}\lim_{x\rightarrow 5} \frac{x^2 - 4} {3x^2 - 2} \end{align*}.

Using Equation (4) of Box #1 (the limit of the quotient is the quotient of the limit),

\begin{align*}\lim_{x \rightarrow 5} \frac{x^2 - 4} {3x^2 - 2} = \frac {\lim_{x \rightarrow 5} (x^2 - 4)} {\lim_{x \rightarrow 5}(3x^2 - 2)}\end{align*}

Making use of the limit of the polynomials theorem, we obtain

\begin{align*}\lim_{x \rightarrow 5} \frac{x^2 - 4} {3x^2 - 2}\end{align*} \begin{align*}= \frac{(5)^2 - 4} {3(5)^2 - 2}\end{align*}
\begin{align*}\mathit = \frac{21} {73}\end{align*}

Example 4

Find the limit: \begin{align*}\lim_{x \to -3} \left( \frac{x^2 + 3x}{x + 3}\right)\end{align*}.

\begin{align*}\left( \frac{x(x + 3)}{x + 3}\right)\end{align*} ..... First factor the numerator

Since both numerator and denominator contain \begin{align*}(x + 3)\end{align*} the function output is the same as the input for any number except where \begin{align*}(x + 3)\end{align*} was undefined, -3.

Since x = y for any number other than -3, as we get closer and closer to -3 on the input, we get closer and closer to -3 on the output (we can get as close as we like, we just can't hit it!).

\begin{align*}\therefore \lim_{x \to -3} \left( \frac{x^2 + 3x}{x + 3}\right) = -3\end{align*}

Example 5


\begin{align*}\lim_{x \to c} f(x) = 7\end{align*}

\begin{align*}\lim_{x \to c} g(x) = 3\end{align*}

Find: \begin{align*}\lim_{x \to c} f(x) g(x)\end{align*}

This example uses rule #3 from above: the limit of the product is the product of the limits. Which means that we need to find the limit of each function, then multiply the limits.

\begin{align*}\lim_{x \to c} f(x) = 7\end{align*}

\begin{align*}\lim_{x \to c} g(x) = 3\end{align*}

Since we are given the limits, we simply multiply \begin{align*}7 \cdot 3 = 21\end{align*}

\begin{align*}\therefore \lim_{x \to c} f(x) g(x) = 21\end{align*}

Example 6


\begin{align*}f(x) = 4x^2 + 3x + 4\end{align*}

\begin{align*}g(x) = 2x^2 -5x -2\end{align*}

Find: \begin{align*}\lim_{x \to -2} f(g(x))\end{align*}

Since these are both continuous functions, we can use \begin{align*}\lim f(g(x)) = f \left(\lim g(x)\right)\end{align*}

\begin{align*}\lim_{x \to -2} 2x^2 -5x -2 = 16\end{align*}

\begin{align*}f(16) = 4(16)^2 + 3(16) + 4 = 1076\end{align*}

\begin{align*}\therefore \lim_{x \to -2} f(g(x)) = 1076\end{align*}


Find the limit:

  1. \begin{align*}\lim_{x \to 2} -5x - 2= \end{align*}
  2. \begin{align*}\lim_{x \to \frac{\pi}{6}} csc (x) = \end{align*}
  3. \begin{align*}\lim_{x \to 256} \sqrt[4]{x}= \end{align*}
  4. \begin{align*}\lim_{x \to0 } \frac{x^2 - 2x}{x}= \end{align*}
  5. \begin{align*}\lim_{x \to 1} 4x = \end{align*}

Find the limit:

  1. \begin{align*}\lim_{x \to -2}\sqrt{-3x + 3} = \end{align*}
  2. Given: \begin{align*}\lim_{x \to c} f(x) = -1 \end{align*} and \begin{align*}\lim_{x \to c } g(x) = -1\end{align*}, find: \begin{align*}\lim_{x \to c} f(x) - g(x) = \end{align*}
  3. Given: \begin{align*}f(x) = -2x^2 + x - 3\end{align*} and \begin{align*}g(x) = -x + 2\end{align*}, find: \begin{align*}\lim_{x \to -5} f(g(x))= \end{align*}
  4. Given: \begin{align*}\lim{x \to c} f(x) = -2 \end{align*} and \begin{align*}\lim_{x \to c} g(x) = -5 \end{align*}, find: \begin{align*}\lim_{x \to c} f(x) + g(x)= \end{align*}
  5. Given: \begin{align*}f(x) = 2x^2 - 2x + 3\end{align*} and \begin{align*}g(x) = -3x^2 - 4x - 5\end{align*}, find: \begin{align*}\lim_{x \to 0} = g(f(x))=\end{align*}
  6. \begin{align*}\lim_{x \to-1} \sqrt{-5x^3 - 5x^2 - 5x - 3} = \end{align*}
  7. Given: \begin{align*}\lim_{x \to c} f(x) = 7 \end{align*} and \begin{align*}\lim_{x \to c} g(x) = -3 \end{align*}, find: \begin{align*}\lim_{x \to c} f(x) - g(x) = \end{align*}
  8. Given: \begin{align*}f(x) = x^2 - 2x + 2\end{align*} and \begin{align*}g(x) = 4x^2 - 2x - 2\end{align*}, find: \begin{align*}\lim_{x \to-1 }f(g(x)) = \end{align*}
  9. \begin{align*}\lim_{x \to 0} \sqrt{-4x^2 + 4x - 3}= \end{align*}
  10. Given: \begin{align*}\lim_{x \to c} f(x) = 5 \end{align*} and \begin{align*}\lim_{x \to c} g(x) = 8\end{align*}, find: \begin{align*}\lim_{x \to c} f(x) + g(x)= \end{align*}
  11. Given: \begin{align*}f(x) = -x^2 -3x -4 \end{align*} and \begin{align*}g(x) = 2x^2 - 5x + 2\end{align*}, find: \begin{align*}\lim_{x \to 4} f(g(x)) = \end{align*}
  12. \begin{align*}\lim_{x \to-6 } \sqrt{-5x - 4} = \end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.4.

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discontinuities The points of discontinuity for a function are the input values of the function where the function is discontinuous.
limit A limit is the value that the output of a function approaches as the input of the function approaches a given value.
limit theorems Limit theorems are a series of statements describing the effects of various mathematical operations on limits.
Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
theorem A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.

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