Calculus Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

Add $x + 1$ and $0$ .

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Simplify terms.

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Step 4.2.1

Simplify each term.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Rewrite the expression.

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

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

Factor $x$ out of $x^{1}$ .

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

Step 4.2.2.2.3

Cancel the common factors.

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Step 4.2.2.2.3.1

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

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

Step 4.2.2.2.3.2

Cancel the common factor.

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

Step 4.2.2.2.3.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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

Step 6

Evaluate the limit.

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Step 6.1

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

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

Step 6.2

Move the limit under the radical sign.

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

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

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

Step 8

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

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

Step 9

Evaluate the limit.

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Step 9.1

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

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

Step 9.2

Simplify the answer.

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Step 9.2.1

Add $1$ and $0$ .

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

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Add $1 + 0$ and $0$ .

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

Step 9.2.2.2

Add $1$ and $0$ .

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

Step 9.2.2.3

Any root of $1$ is $1$ .

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

Step 9.2.2.4

Multiply $- 1$ by $1$ .

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

Step 9.2.2.5

Multiply $- 1$ by $1$ .

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

Step 9.2.2.6

Subtract $1$ from $- 1$ .

$\frac{1}{- 2}$

Step 9.2.3

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$