Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Expand $(x - 1) (2 x^{2} + x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.2

Combine the opposite terms in $x (2 x^{2}) + x \cdot x + x \cdot 1 - 1 (2 x^{2}) - 1 x - 1 \cdot 1$ .

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Step 1.2.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 1.2.2

Subtract $1 x$ from $1 x$ .

$lim_{x \to \infty} \frac{{(1 - 2 x)}^{3}}{x (2 x^{2}) + x \cdot x - 1 (2 x^{2}) + 0 - 1 \cdot 1}$

Step 1.2.3

Add $x (2 x^{2}) + x \cdot x - 1 (2 x^{2})$ and $0$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

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

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Step 1.3.2.1

Move $x^{2}$ .

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

Step 1.3.2.2

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

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Step 1.3.2.2.1

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

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

Step 1.3.2.2.2

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

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

Step 1.3.2.3

Add $2$ and $1$ .

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

Step 1.3.3

Multiply $x$ by $x$ .

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

Step 1.3.4

Multiply $2$ by $- 1$ .

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

Step 1.3.5

Multiply $- 1$ by $1$ .

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

Step 1.4

Subtract $2 x^{2}$ from $x^{2}$ .

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

Step 2

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

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

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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

Step 3.2

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

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

Step 3.3

Move the exponent $3$ from ${(\frac{1}{x} + \frac{- 2 x}{x})}^{3}$ outside the limit using the Limits Power Rule.

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

Step 3.4

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

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

Step 4

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

$\frac{{(0 + lim_{x \to \infty} \frac{- 2 x}{x})}^{3}}{lim_{x \to \infty} 2 - \frac{1}{x} - \frac{1}{x^{3}}}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $- 2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(0 - 2 lim_{x \to \infty} \frac{x}{x})}^{3}}{lim_{x \to \infty} 2 - \frac{1}{x} - \frac{1}{x^{3}}}$

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

$\frac{{(0 - 2 lim_{x \to \infty} \frac{x}{x})}^{3}}{lim_{x \to \infty} 2 - \frac{1}{x} - \frac{1}{x^{3}}}$

Step 5.2.2

Rewrite the expression.

$\frac{{(0 - 2 lim_{x \to \infty} 1)}^{3}}{lim_{x \to \infty} 2 - \frac{1}{x} - \frac{1}{x^{3}}}$

Step 5.3

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

$\frac{{(0 - 2 \cdot 1)}^{3}}{lim_{x \to \infty} 2 - \frac{1}{x} - \frac{1}{x^{3}}}$

Step 5.4

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

$\frac{{(0 - 2 \cdot 1)}^{3}}{lim_{x \to \infty} 2 - lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 5.5

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{{(0 - 2 \cdot 1)}^{3}}{2 - lim_{x \to \infty} \frac{1}{x} - lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 6

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

$\frac{{(0 - 2 \cdot 1)}^{3}}{2 - 0 - lim_{x \to \infty} \frac{1}{x^{3}}}$

Step 7

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

$\frac{{(0 - 2 \cdot 1)}^{3}}{2 - 0 - 0}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $- 2$ by $1$ .

$\frac{{(0 - 2)}^{3}}{2 - 0 - 0}$

Step 8.1.2

Subtract $2$ from $0$ .

$\frac{{(- 2)}^{3}}{2 - 0 - 0}$

Step 8.1.3

Raise $- 2$ to the power of $3$ .

$\frac{- 8}{2 - 0 - 0}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $- 1$ by $0$ .

$\frac{- 8}{2 + 0 - 0}$

Step 8.2.2

Multiply $- 1$ by $0$ .

$\frac{- 8}{2 + 0 + 0}$

Step 8.2.3

Add $2 + 0$ and $0$ .

$\frac{- 8}{2 + 0}$

Step 8.2.4

Add $2$ and $0$ .

$\frac{- 8}{2}$

Step 8.3

Divide $- 8$ by $2$ .

$- 4$