Calculus Examples

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

Step 1

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

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Step 1.1

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

$lim_{x \to - \infty} \frac{x}{\sqrt{x \cdot x - x}}$

Step 1.2

Factor $x$ out of $- x$ .

$lim_{x \to - \infty} \frac{x}{\sqrt{x \cdot x + x \cdot - 1}}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$lim_{x \to - \infty} \frac{x}{\sqrt{x (x - 1)}}$

Step 2

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

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

Step 3

Cancel the common factor of $x$ .

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

Step 4

Cancel the common factors.

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Step 4.1

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

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (x - 1)}{x \cdot x}}}$

Step 4.2

Cancel the common factor.

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x (x - 1)}{x \cdot x}}}$

Step 4.3

Rewrite the expression.

$lim_{x \to - \infty} \frac{1}{- \sqrt{\frac{x - 1}{x}}}$

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 $- \infty$ .

$\frac{lim_{x \to - \infty} 1}{lim_{x \to - \infty} - \sqrt{\frac{x - 1}{x}}}$

Step 5.2

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

$\frac{1}{lim_{x \to - \infty} - \sqrt{\frac{x - 1}{x}}}$

Step 5.3

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

$\frac{1}{- lim_{x \to - \infty} \sqrt{\frac{x - 1}{x}}}$

Step 5.4

Move the limit under the radical sign.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{x - 1}{x}}}$

Step 6

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

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{\frac{x}{x} - \frac{1}{x}}{\frac{x}{x}}}}$

Step 7

Evaluate the limit.

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Step 7.1

Cancel the common factor of $x$ .

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Step 7.1.1

Cancel the common factor.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{\frac{x}{x} - \frac{1}{x}}{\frac{x}{x}}}}$

Step 7.1.2

Rewrite the expression.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{1}{x}}{\frac{x}{x}}}}$

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{1}{x}}{\frac{x}{x}}}}$

Step 7.2.2

Rewrite the expression.

$\frac{1}{- \sqrt{lim_{x \to - \infty} \frac{1 - \frac{1}{x}}{1}}}$

Step 7.3

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

$\frac{1}{- \sqrt{\frac{lim_{x \to - \infty} 1 - \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 7.4

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

$\frac{1}{- \sqrt{\frac{lim_{x \to - \infty} 1 - lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 7.5

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

$\frac{1}{- \sqrt{\frac{1 - lim_{x \to - \infty} \frac{1}{x}}{lim_{x \to - \infty} 1}}}$

Step 8

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

$\frac{1}{- \sqrt{\frac{1 - 0}{lim_{x \to - \infty} 1}}}$

Step 9

Evaluate the limit.

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Step 9.1

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

$\frac{1}{- \sqrt{\frac{1 - 0}{1}}}$

Step 9.2

Simplify the answer.

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Step 9.2.1

Divide $1 - 0$ by $1$ .

$\frac{1}{- \sqrt{1 - 0}}$

Step 9.2.2

Cancel the common factor of $1$ and $- 1$ .

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Step 9.2.2.1

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

$\frac{- 1 (- 1)}{- \sqrt{1 - 0}}$

Step 9.2.2.2

Move the negative in front of the fraction.

$- \frac{1}{\sqrt{1 - 0}}$

Step 9.2.3

Simplify the denominator.

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Step 9.2.3.1

Multiply $- 1$ by $0$ .

$- \frac{1}{\sqrt{1 + 0}}$

Step 9.2.3.2

Add $1$ and $0$ .

$- \frac{1}{\sqrt{1}}$

Step 9.2.3.3

Any root of $1$ is $1$ .

$- \frac{1}{1}$

Step 9.2.4

Cancel the common factor of $1$ .

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Step 9.2.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 9.2.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 9.2.5

Multiply $- 1$ by $1$ .

$- 1$