Calculus Examples

$lim_{x \to \infty} \frac{(x - 2) (3 x^{2} + 3)}{{(6 x + 4)}^{3}}$

Step 1

Simplify.

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Step 1.1

Expand $(x - 2) (3 x^{2} + 3)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$lim_{x \to \infty} \frac{x (3 x^{2} + 3) - 2 (3 x^{2} + 3)}{{(6 x + 4)}^{3}}$

Step 1.1.2

Apply the distributive property.

$lim_{x \to \infty} \frac{x (3 x^{2}) + x \cdot 3 - 2 (3 x^{2} + 3)}{{(6 x + 4)}^{3}}$

Step 1.1.3

Apply the distributive property.

$lim_{x \to \infty} \frac{x (3 x^{2}) + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$lim_{x \to \infty} \frac{3 x \cdot x^{2} + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.2

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

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Step 1.2.2.1

Move $x^{2}$ .

$lim_{x \to \infty} \frac{3 (x^{2} x) + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$lim_{x \to \infty} \frac{3 (x^{2} x^{1}) + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.2.2.2

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

$lim_{x \to \infty} \frac{3 x^{2 + 1} + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.2.3

Add $2$ and $1$ .

$lim_{x \to \infty} \frac{3 x^{3} + x \cdot 3 - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.3

Move $3$ to the left of $x$ .

$lim_{x \to \infty} \frac{3 x^{3} + 3 \cdot x - 2 (3 x^{2}) - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.4

Multiply $3$ by $- 2$ .

$lim_{x \to \infty} \frac{3 x^{3} + 3 x - 6 x^{2} - 2 \cdot 3}{{(6 x + 4)}^{3}}$

Step 1.2.5

Multiply $- 2$ by $3$ .

$lim_{x \to \infty} \frac{3 x^{3} + 3 x - 6 x^{2} - 6}{{(6 x + 4)}^{3}}$

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator.

$lim_{x \to \infty} \frac{\frac{3 x^{3}}{x^{3}} + \frac{3 x}{x^{3}} + \frac{- 6 x^{2}}{x^{3}} + \frac{- 6}{x^{3}}}{{(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

$lim_{x \to \infty} \frac{3 + \frac{3}{x^{2}} - \frac{6}{x} - \frac{6}{x^{3}}}{{(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 3 + \frac{3}{x^{2}} - \frac{6}{x} - \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{lim_{x \to \infty} 3 + lim_{x \to \infty} \frac{3}{x^{2}} - lim_{x \to \infty} \frac{6}{x} - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 3.4

Evaluate the limit of $3$ which is constant as $x$ approaches $\infty$ .

$\frac{3 + lim_{x \to \infty} \frac{3}{x^{2}} - lim_{x \to \infty} \frac{6}{x} - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 3.5

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 3 lim_{x \to \infty} \frac{1}{x^{2}} - lim_{x \to \infty} \frac{6}{x} - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{2}}$ approaches $0$ .

$\frac{3 + 3 \cdot 0 - lim_{x \to \infty} \frac{6}{x} - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 5

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 3 \cdot 0 - 6 lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - lim_{x \to \infty} \frac{6}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 7

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 lim_{x \to \infty} \frac{1}{x^{3}}}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 8

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x^{3}}$ approaches $0$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{lim_{x \to \infty} {(\frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 9

Evaluate the limit.

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Step 9.1

Move the exponent $3$ from ${(\frac{6 x}{x} + \frac{4}{x})}^{3}$ outside the limit using the Limits Power Rule.

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(lim_{x \to \infty} \frac{6 x}{x} + \frac{4}{x})}^{3}}$

Step 9.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\infty$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(lim_{x \to \infty} \frac{6 x}{x} + lim_{x \to \infty} \frac{4}{x})}^{3}}$

Step 9.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 lim_{x \to \infty} \frac{x}{x} + lim_{x \to \infty} \frac{4}{x})}^{3}}$

Step 9.4

Cancel the common factor of $x$ .

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Step 9.4.1

Cancel the common factor.

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 lim_{x \to \infty} \frac{x}{x} + lim_{x \to \infty} \frac{4}{x})}^{3}}$

Step 9.4.2

Rewrite the expression.

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 lim_{x \to \infty} 1 + lim_{x \to \infty} \frac{4}{x})}^{3}}$

Step 9.5

Evaluate the limit of $1$ which is constant as $x$ approaches $\infty$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 \cdot 1 + lim_{x \to \infty} \frac{4}{x})}^{3}}$

Step 9.6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 \cdot 1 + 4 lim_{x \to \infty} \frac{1}{x})}^{3}}$

Step 10

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{3 + 3 \cdot 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply $3$ by $0$ .

$\frac{3 + 0 - 6 \cdot 0 - 6 \cdot 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.1.2

Multiply $- 6$ by $0$ .

$\frac{3 + 0 + 0 - 6 \cdot 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.1.3

Multiply $- 6$ by $0$ .

$\frac{3 + 0 + 0 + 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.1.4

Add $3 + 0 + 0$ and $0$ .

$\frac{3 + 0 + 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.1.5

Add $3 + 0$ and $0$ .

$\frac{3 + 0}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.1.6

Add $3$ and $0$ .

$\frac{3}{{(6 \cdot 1 + 4 \cdot 0)}^{3}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Multiply $6$ by $1$ .

$\frac{3}{{(6 + 4 \cdot 0)}^{3}}$

Step 11.2.2

Multiply $4$ by $0$ .

$\frac{3}{{(6 + 0)}^{3}}$

Step 11.2.3

Add $6$ and $0$ .

$\frac{3}{6^{3}}$

Step 11.2.4

Raise $6$ to the power of $3$ .

$\frac{3}{216}$

Step 11.3

Cancel the common factor of $3$ and $216$ .

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Step 11.3.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{216}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $3$ out of $216$ .

$\frac{3 \cdot 1}{3 \cdot 72}$

Step 11.3.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 72}$

Step 11.3.2.3

Rewrite the expression.

$\frac{1}{72}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{72}$

Decimal Form:

$0.013\overline{8}$