Calculus Examples

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

Step 1

Factor $x$ out of $4 x^{4} - x$ .

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Step 1.1

Factor $x$ out of $4 x^{4}$ .

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

Step 1.2

Factor $x$ out of $- x$ .

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

Step 1.3

Factor $x$ out of $x (4 x^{3}) + x \cdot - 1$ .

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

Step 2

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $- \sqrt{x^{4}} = x^{2}$ .

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

Step 3

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

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

Step 4

Cancel the common factors.

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Step 4.1

Factor $x$ out of $x^{4}$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 5.3

Move the limit under the radical sign.

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

Step 6

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

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

Step 7

Evaluate the limit.

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Step 7.1

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

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Divide $4$ by $1$ .

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

Step 7.2

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

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 8

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

$\frac{- \sqrt{\frac{4 - 0}{lim_{x \to - \infty} 1}}}{lim_{x \to - \infty} 2 + \frac{3}{x^{2}}}$

Step 9

Evaluate the limit.

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Step 9.1

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

$\frac{- \sqrt{\frac{4 - 0}{1}}}{lim_{x \to - \infty} 2 + \frac{3}{x^{2}}}$

Step 9.2

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

$\frac{- \sqrt{\frac{4 - 0}{1}}}{lim_{x \to - \infty} 2 + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 9.3

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

$\frac{- \sqrt{\frac{4 - 0}{1}}}{2 + lim_{x \to - \infty} \frac{3}{x^{2}}}$

Step 9.4

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

$\frac{- \sqrt{\frac{4 - 0}{1}}}{2 + 3 lim_{x \to - \infty} \frac{1}{x^{2}}}$

Step 10

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

$\frac{- \sqrt{\frac{4 - 0}{1}}}{2 + 3 \cdot 0}$

Step 11

Simplify the answer.

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Step 11.1

Divide $4 - 0$ by $1$ .

$\frac{- \sqrt{4 - 0}}{2 + 3 \cdot 0}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Multiply $- 1$ by $0$ .

$\frac{- \sqrt{4 + 0}}{2 + 3 \cdot 0}$

Step 11.2.2

Add $4$ and $0$ .

$\frac{- \sqrt{4}}{2 + 3 \cdot 0}$

Step 11.2.3

Rewrite $4$ as $2^{2}$ .

$\frac{- \sqrt{2^{2}}}{2 + 3 \cdot 0}$

Step 11.2.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{- 1 \cdot 2}{2 + 3 \cdot 0}$

Step 11.3

Simplify the denominator.

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Step 11.3.1

Multiply $3$ by $0$ .

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

Step 11.3.2

Add $2$ and $0$ .

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

Step 11.4

Multiply $- 1$ by $2$ .

$\frac{- 2}{2}$

Step 11.5

Divide $- 2$ by $2$ .

$- 1$