Precalculus Examples

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

Step 1

Cancel the common factors.

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Step 1.1

Factor $x^{2}$ out of $x^{4} (2 x + 1)$ .

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

Step 1.2

Cancel the common factor.

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

Step 1.3

Rewrite the expression.

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

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $x^{2}$ by $1$ .

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

Step 2.4

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

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Step 2.4.1

Move $x$ .

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

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.3

Add $1$ and $2$ .

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

Step 3

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x^{3}$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $x^{3}$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Divide $- 4$ by $1$ .

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

Step 4.2

Simplify each term.

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Step 4.2.1

Cancel the common factor of $x^{3}$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $2$ by $1$ .

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

Step 4.2.2

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 4.2.2.1

Multiply by $1$ .

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

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

Cancel the common factor.

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

Step 4.2.2.2.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

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

$\frac{0 - lim_{x \to \infty} 4}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 6

Evaluate the limit.

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Step 6.1

Evaluate the limit of $4$ which is constant as $x$ approaches $\infty$ .

$\frac{0 - 1 \cdot 4}{lim_{x \to \infty} 2 + \frac{1}{x}}$

Step 6.2

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

$\frac{0 - 1 \cdot 4}{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{1}{x}}$

Step 6.3

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

$\frac{0 - 1 \cdot 4}{2 + lim_{x \to \infty} \frac{1}{x}}$

Step 7

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

$\frac{0 - 1 \cdot 4}{2 + 0}$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $0 - 1 \cdot 4$ and $2 + 0$ .

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Step 8.1.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{- 1 (0) - 1 \cdot 4}{2 + 0}$

Step 8.1.2

Factor $- 1$ out of $- 1 \cdot 4$ .

$\frac{- 1 (0) - 1 (4)}{2 + 0}$

Step 8.1.3

Factor $- 1$ out of $- 1 (0) - 1 (4)$ .

$\frac{- 1 (0 + 4)}{2 + 0}$

Step 8.1.4

Factor $2$ out of $- 1 (0 + 4)$ .

$\frac{2 (- 1 (0 + 2))}{2 + 0}$

Step 8.1.5

Cancel the common factors.

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Step 8.1.5.1

Factor $2$ out of $2$ .

$\frac{2 (- 1 (0 + 2))}{2 (1) + 0}$

Step 8.1.5.2

Factor $2$ out of $0$ .

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

Step 8.1.5.3

Factor $2$ out of $2 \cdot 1 + 2 \cdot 0$ .

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

Step 8.1.5.4

Cancel the common factor.

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

Step 8.1.5.5

Rewrite the expression.

$\frac{- 1 (0 + 2)}{1 + 0}$

Step 8.2

Add $0$ and $2$ .

$\frac{- 1 \cdot 2}{1 + 0}$

Step 8.3

Add $1$ and $0$ .

$\frac{- 1 \cdot 2}{1}$

Step 8.4

Multiply $- 1$ by $2$ .

$\frac{- 2}{1}$

Step 8.5

Divide $- 2$ by $1$ .

$- 2$