Calculus Examples

$lim_{x \to 3} v^{2} (3 v - 4) (9 - v^{2})$

Step 1

Evaluate the limit of $v^{2} (3 v - 4) (9 - v^{2})$ which is constant as $x$ approaches $3$ .

$v^{2} (3 v - 4) (9 - v^{2})$

Step 2

Simplify the answer.

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Step 2.1

Apply the distributive property.

$(v^{2} (3 v) + v^{2} \cdot - 4) (9 - v^{2})$

Step 2.2

Rewrite using the commutative property of multiplication.

$(3 v^{2} v + v^{2} \cdot - 4) (9 - v^{2})$

Step 2.3

Move $- 4$ to the left of $v^{2}$ .

$(3 v^{2} v - 4 \cdot v^{2}) (9 - v^{2})$

Step 2.4

Multiply $v^{2}$ by $v$ by adding the exponents.

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Step 2.4.1

Move $v$ .

$(3 (v \cdot v^{2}) - 4 \cdot v^{2}) (9 - v^{2})$

Step 2.4.2

Multiply $v$ by $v^{2}$ .

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Step 2.4.2.1

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

$(3 (v^{1} v^{2}) - 4 \cdot v^{2}) (9 - v^{2})$

Step 2.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(3 v^{1 + 2} - 4 \cdot v^{2}) (9 - v^{2})$

Step 2.4.3

Add $1$ and $2$ .

$(3 v^{3} - 4 \cdot v^{2}) (9 - v^{2})$

$(3 v^{3} - 4 v^{2}) (9 - v^{2})$

Step 2.5

Expand $(3 v^{3} - 4 v^{2}) (9 - v^{2})$ using the FOIL Method.

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Step 2.5.1

Apply the distributive property.

$3 v^{3} (9 - v^{2}) - 4 v^{2} (9 - v^{2})$

Step 2.5.2

Apply the distributive property.

$3 v^{3} \cdot 9 + 3 v^{3} (- v^{2}) - 4 v^{2} (9 - v^{2})$

Step 2.5.3

Apply the distributive property.

$3 v^{3} \cdot 9 + 3 v^{3} (- v^{2}) - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6

Simplify each term.

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Step 2.6.1

Multiply $9$ by $3$ .

$27 v^{3} + 3 v^{3} (- v^{2}) - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.2

Rewrite using the commutative property of multiplication.

$27 v^{3} + 3 \cdot - 1 v^{3} v^{2} - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.3

Multiply $v^{3}$ by $v^{2}$ by adding the exponents.

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Step 2.6.3.1

Move $v^{2}$ .

$27 v^{3} + 3 \cdot - 1 (v^{2} v^{3}) - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 v^{3} + 3 \cdot - 1 v^{2 + 3} - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.3.3

Add $2$ and $3$ .

$27 v^{3} + 3 \cdot - 1 v^{5} - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.4

Multiply $3$ by $- 1$ .

$27 v^{3} - 3 v^{5} - 4 v^{2} \cdot 9 - 4 v^{2} (- v^{2})$

Step 2.6.5

Multiply $9$ by $- 4$ .

$27 v^{3} - 3 v^{5} - 36 v^{2} - 4 v^{2} (- v^{2})$

Step 2.6.6

Rewrite using the commutative property of multiplication.

$27 v^{3} - 3 v^{5} - 36 v^{2} - 4 \cdot - 1 v^{2} v^{2}$

Step 2.6.7

Multiply $v^{2}$ by $v^{2}$ by adding the exponents.

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Step 2.6.7.1

Move $v^{2}$ .

$27 v^{3} - 3 v^{5} - 36 v^{2} - 4 \cdot - 1 (v^{2} v^{2})$

Step 2.6.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 v^{3} - 3 v^{5} - 36 v^{2} - 4 \cdot - 1 v^{2 + 2}$

Step 2.6.7.3

Add $2$ and $2$ .

$27 v^{3} - 3 v^{5} - 36 v^{2} - 4 \cdot - 1 v^{4}$

Step 2.6.8

Multiply $- 4$ by $- 1$ .

$27 v^{3} - 3 v^{5} - 36 v^{2} + 4 v^{4}$