Calculus Examples

$lim_{x \to 8} 729 - 729 \frac{2 n^{3} + 3 n^{2} + n}{6 n^{3}}$

Step 1

Evaluate the limit of $729 - 729 \frac{2 n^{3} + 3 n^{2} + n}{6 n^{3}}$ which is constant as $x$ approaches $8$ .

$729 - 729 \frac{2 n^{3} + 3 n^{2} + n}{6 n^{3}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the numerator.

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Step 2.1.1.1

Factor $n$ out of $2 n^{3} + 3 n^{2} + n$ .

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Step 2.1.1.1.1

Factor $n$ out of $2 n^{3}$ .

$729 - 729 \frac{n (2 n^{2}) + 3 n^{2} + n}{6 n^{3}}$

Step 2.1.1.1.2

Factor $n$ out of $3 n^{2}$ .

$729 - 729 \frac{n (2 n^{2}) + n (3 n) + n}{6 n^{3}}$

Step 2.1.1.1.3

Raise $n$ to the power of $1$ .

$729 - 729 \frac{n (2 n^{2}) + n (3 n) + n^{1}}{6 n^{3}}$

Step 2.1.1.1.4

Factor $n$ out of $n^{1}$ .

$729 - 729 \frac{n (2 n^{2}) + n (3 n) + n \cdot 1}{6 n^{3}}$

Step 2.1.1.1.5

Factor $n$ out of $n (2 n^{2}) + n (3 n)$ .

$729 - 729 \frac{n (2 n^{2} + 3 n) + n \cdot 1}{6 n^{3}}$

Step 2.1.1.1.6

Factor $n$ out of $n (2 n^{2} + 3 n) + n \cdot 1$ .

$729 - 729 \frac{n (2 n^{2} + 3 n + 1)}{6 n^{3}}$

Step 2.1.1.2

Factor by grouping.

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Step 2.1.1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

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Step 2.1.1.2.1.1

Factor $3$ out of $3 n$ .

$729 - 729 \frac{n (2 n^{2} + 3 (n) + 1)}{6 n^{3}}$

Step 2.1.1.2.1.2

Rewrite $3$ as $1$ plus $2$

$729 - 729 \frac{n (2 n^{2} + (1 + 2) n + 1)}{6 n^{3}}$

Step 2.1.1.2.1.3

Apply the distributive property.

$729 - 729 \frac{n (2 n^{2} + 1 n + 2 n + 1)}{6 n^{3}}$

Step 2.1.1.2.1.4

Multiply $n$ by $1$ .

$729 - 729 \frac{n (2 n^{2} + n + 2 n + 1)}{6 n^{3}}$

Step 2.1.1.2.2

Factor out the greatest common factor from each group.

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Step 2.1.1.2.2.1

Group the first two terms and the last two terms.

$729 - 729 \frac{n ((2 n^{2} + n) + 2 n + 1)}{6 n^{3}}$

Step 2.1.1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$729 - 729 \frac{n (n (2 n + 1) + 1 (2 n + 1))}{6 n^{3}}$

Step 2.1.1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 n + 1$ .

$729 - 729 \frac{n ((2 n + 1) (n + 1))}{6 n^{3}}$

$729 - 729 \frac{n (2 n + 1) (n + 1)}{6 n^{3}}$

Step 2.1.2

Cancel the common factor of $3$ .

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Step 2.1.2.1

Factor $3$ out of $- 729$ .

$729 + 3 (- 243) \frac{n (2 n + 1) (n + 1)}{6 n^{3}}$

Step 2.1.2.2

Factor $3$ out of $6 n^{3}$ .

$729 + 3 (- 243) \frac{n (2 n + 1) (n + 1)}{3 (2 n^{3})}$

Step 2.1.2.3

Cancel the common factor.

$729 + 3 \cdot - 243 \frac{n (2 n + 1) (n + 1)}{3 (2 n^{3})}$

Step 2.1.2.4

Rewrite the expression.

$729 - 243 \frac{n (2 n + 1) (n + 1)}{2 n^{3}}$

Step 2.1.3

Combine $- 243$ and $\frac{n (2 n + 1) (n + 1)}{2 n^{3}}$ .

$729 + \frac{- 243 (n (2 n + 1) (n + 1))}{2 n^{3}}$

Step 2.1.4

Cancel the common factor of $n$ and $n^{3}$ .

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Step 2.1.4.1

Factor $n$ out of $- 243 (n (2 n + 1) (n + 1))$ .

$729 + \frac{n (- 243 ((2 n + 1) (n + 1)))}{2 n^{3}}$

Step 2.1.4.2

Cancel the common factors.

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Step 2.1.4.2.1

Factor $n$ out of $2 n^{3}$ .

$729 + \frac{n (- 243 ((2 n + 1) (n + 1)))}{n (2 n^{2})}$

Step 2.1.4.2.2

Cancel the common factor.

$729 + \frac{n (- 243 ((2 n + 1) (n + 1)))}{n (2 n^{2})}$

Step 2.1.4.2.3

Rewrite the expression.

$729 + \frac{- 243 ((2 n + 1) (n + 1))}{2 n^{2}}$

Step 2.1.5

Move the negative in front of the fraction.

$729 - \frac{243 (2 n + 1) (n + 1)}{2 n^{2}}$

Step 2.2

To write $729$ as a fraction with a common denominator, multiply by $\frac{2 n^{2}}{2 n^{2}}$ .

$729 \cdot \frac{2 n^{2}}{2 n^{2}} - \frac{243 (2 n + 1) (n + 1)}{2 n^{2}}$

Step 2.3

Combine $729$ and $\frac{2 n^{2}}{2 n^{2}}$ .

$\frac{729 (2 n^{2})}{2 n^{2}} - \frac{243 (2 n + 1) (n + 1)}{2 n^{2}}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{729 (2 n^{2}) - 243 (2 n + 1) (n + 1)}{2 n^{2}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $243$ out of $729 \cdot 2 n^{2} - 243 (2 n + 1) (n + 1)$ .

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Step 2.5.1.1

Factor $243$ out of $729 \cdot 2 n^{2}$ .

$\frac{243 (3 \cdot 2 n^{2}) - 243 (2 n + 1) (n + 1)}{2 n^{2}}$

Step 2.5.1.2

Factor $243$ out of $- 243 (2 n + 1) (n + 1)$ .

$\frac{243 (3 \cdot 2 n^{2}) + 243 (- (2 n + 1) (n + 1))}{2 n^{2}}$

Step 2.5.1.3

Factor $243$ out of $243 (3 \cdot 2 n^{2}) + 243 (- (2 n + 1) (n + 1))$ .

$\frac{243 (3 \cdot 2 n^{2} - (2 n + 1) (n + 1))}{2 n^{2}}$

Step 2.5.2

Multiply $3$ by $2$ .

$\frac{243 (6 n^{2} - (2 n + 1) (n + 1))}{2 n^{2}}$

Step 2.5.3

Apply the distributive property.

$\frac{243 (6 n^{2} + (- (2 n) - 1 \cdot 1) (n + 1))}{2 n^{2}}$

Step 2.5.4

Multiply $2$ by $- 1$ .

$\frac{243 (6 n^{2} + (- 2 n - 1 \cdot 1) (n + 1))}{2 n^{2}}$

Step 2.5.5

Multiply $- 1$ by $1$ .

$\frac{243 (6 n^{2} + (- 2 n - 1) (n + 1))}{2 n^{2}}$

Step 2.5.6

Expand $(- 2 n - 1) (n + 1)$ using the FOIL Method.

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Step 2.5.6.1

Apply the distributive property.

$\frac{243 (6 n^{2} - 2 n (n + 1) - 1 (n + 1))}{2 n^{2}}$

Step 2.5.6.2

Apply the distributive property.

$\frac{243 (6 n^{2} - 2 n \cdot n - 2 n \cdot 1 - 1 (n + 1))}{2 n^{2}}$

Step 2.5.6.3

Apply the distributive property.

$\frac{243 (6 n^{2} - 2 n \cdot n - 2 n \cdot 1 - 1 n - 1 \cdot 1)}{2 n^{2}}$

Step 2.5.7

Simplify and combine like terms.

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Step 2.5.7.1

Simplify each term.

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Step 2.5.7.1.1

Multiply $n$ by $n$ by adding the exponents.

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Step 2.5.7.1.1.1

Move $n$ .

$\frac{243 (6 n^{2} - 2 (n \cdot n) - 2 n \cdot 1 - 1 n - 1 \cdot 1)}{2 n^{2}}$

Step 2.5.7.1.1.2

Multiply $n$ by $n$ .

$\frac{243 (6 n^{2} - 2 n^{2} - 2 n \cdot 1 - 1 n - 1 \cdot 1)}{2 n^{2}}$

Step 2.5.7.1.2

Multiply $- 2$ by $1$ .

$\frac{243 (6 n^{2} - 2 n^{2} - 2 n - 1 n - 1 \cdot 1)}{2 n^{2}}$

Step 2.5.7.1.3

Rewrite $- 1 n$ as $- n$ .

$\frac{243 (6 n^{2} - 2 n^{2} - 2 n - n - 1 \cdot 1)}{2 n^{2}}$

Step 2.5.7.1.4

Multiply $- 1$ by $1$ .

$\frac{243 (6 n^{2} - 2 n^{2} - 2 n - n - 1)}{2 n^{2}}$

Step 2.5.7.2

Subtract $n$ from $- 2 n$ .

$\frac{243 (6 n^{2} - 2 n^{2} - 3 n - 1)}{2 n^{2}}$

Step 2.5.8

Subtract $2 n^{2}$ from $6 n^{2}$ .

$\frac{243 (4 n^{2} - 3 n - 1)}{2 n^{2}}$

Step 2.5.9

Factor by grouping.

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Step 2.5.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = - 3$ .

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Step 2.5.9.1.1

Factor $- 3$ out of $- 3 n$ .

$\frac{243 (4 n^{2} - 3 (n) - 1)}{2 n^{2}}$

Step 2.5.9.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$\frac{243 (4 n^{2} + (1 - 4) n - 1)}{2 n^{2}}$

Step 2.5.9.1.3

Apply the distributive property.

$\frac{243 (4 n^{2} + 1 n - 4 n - 1)}{2 n^{2}}$

Step 2.5.9.1.4

Multiply $n$ by $1$ .

$\frac{243 (4 n^{2} + n - 4 n - 1)}{2 n^{2}}$

Step 2.5.9.2

Factor out the greatest common factor from each group.

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Step 2.5.9.2.1

Group the first two terms and the last two terms.

$\frac{243 ((4 n^{2} + n) - 4 n - 1)}{2 n^{2}}$

Step 2.5.9.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{243 (n (4 n + 1) - (4 n + 1))}{2 n^{2}}$

Step 2.5.9.3

Factor the polynomial by factoring out the greatest common factor, $4 n + 1$ .

$\frac{243 ((4 n + 1) (n - 1))}{2 n^{2}}$

$\frac{243 (4 n + 1) (n - 1)}{2 n^{2}}$