Calculus Examples

$lim_{x \to \infty} \sqrt{x^{4} - 5 x^{2}} - x^{2}$

Step 1

Multiply to rationalize the numerator.

$lim_{x \to \infty} \frac{(\sqrt{x^{4} - 5 x^{2}} - x^{2}) (\sqrt{x^{4} - 5 x^{2}} + x^{2})}{\sqrt{x^{4} - 5 x^{2}} + x^{2}}$

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

$lim_{x \to \infty} \frac{{\sqrt{x^{4} - 5 x^{2}}}^{2} + \sqrt{x^{4} - 5 x^{2}} x^{2} + \sqrt{x^{4} - 5 x^{2}} (- x^{2}) - x^{4}}{\sqrt{x^{4} - 5 x^{2}} + x^{2}}$

Step 2.2

Simplify.

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Step 2.2.1

Subtract $x^{4}$ from $x^{4}$ .

$lim_{x \to \infty} \frac{- 5 x^{2} + 0}{\sqrt{x^{4} - 5 x^{2}} + x^{2}}$

Step 2.2.2

Add $- 5 x^{2}$ and $0$ .

$lim_{x \to \infty} \frac{- 5 x^{2}}{\sqrt{x^{4} - 5 x^{2}} + x^{2}}$

Step 3

Evaluate the limit.

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Step 3.1

Simplify each term.

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Step 3.1.1

Factor $x^{2}$ out of $x^{4} - 5 x^{2}$ .

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Step 3.1.1.1

Factor $x^{2}$ out of $x^{4}$ .

$lim_{x \to \infty} \frac{- 5 x^{2}}{\sqrt{x^{2} x^{2} - 5 x^{2}} + x^{2}}$

Step 3.1.1.2

Factor $x^{2}$ out of $- 5 x^{2}$ .

$lim_{x \to \infty} \frac{- 5 x^{2}}{\sqrt{x^{2} x^{2} + x^{2} \cdot - 5} + x^{2}}$

Step 3.1.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 5$ .

$lim_{x \to \infty} \frac{- 5 x^{2}}{\sqrt{x^{2} (x^{2} - 5)} + x^{2}}$

Step 3.1.2

Pull terms out from under the radical.

$lim_{x \to \infty} \frac{- 5 x^{2}}{x \sqrt{x^{2} - 5} + x^{2}}$

Step 3.2

Move the term $- 5$ outside of the limit because it is constant with respect to $x$ .

$- 5 lim_{x \to \infty} \frac{x^{2}}{x \sqrt{x^{2} - 5} + x^{2}}$

Step 4

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

$- 5 lim_{x \to \infty} \frac{\frac{x^{2}}{x^{2}}}{\frac{x \sqrt{x^{2} - 5}}{x^{2}} + \frac{x^{2}}{x^{2}}}$

Step 5

Evaluate the limit.

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Step 5.1

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

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Step 5.1.1

Cancel the common factor.

$- 5 lim_{x \to \infty} \frac{\frac{x^{2}}{x^{2}}}{\frac{x \sqrt{x^{2} - 5}}{x^{2}} + \frac{x^{2}}{x^{2}}}$

Step 5.1.2

Rewrite the expression.

$- 5 lim_{x \to \infty} \frac{1}{\frac{x \sqrt{x^{2} - 5}}{x^{2}} + \frac{x^{2}}{x^{2}}}$

Step 5.2

Simplify each term.

$- 5 lim_{x \to \infty} \frac{1}{\frac{\sqrt{x^{2} - 5}}{x} + 1}$

Step 5.3

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

$- 5 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} \frac{\sqrt{x^{2} - 5}}{x} + 1}$

Step 5.4

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x^{2} - 5}}{x} + 1}$

Step 5.5

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{x^{2} - 5}}{x} + lim_{x \to \infty} 1}$

Step 6

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

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{\frac{x}{x}} + lim_{x \to \infty} 1}$

Step 7

Evaluate the limit.

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Step 7.1

Cancel the common factor of $x$ .

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{1} + lim_{x \to \infty} 1}$

Step 7.2

Simplify each term.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{\frac{x^{2}}{x^{2}} + \frac{- 5}{x^{2}}}}{1} + lim_{x \to \infty} 1}$

Step 7.2.1.2

Rewrite the expression.

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{1 + \frac{- 5}{x^{2}}}}{1} + lim_{x \to \infty} 1}$

Step 7.2.2

Move the negative in front of the fraction.

$- 5 \frac{1}{lim_{x \to \infty} \frac{\sqrt{1 - \frac{5}{x^{2}}}}{1} + lim_{x \to \infty} 1}$

Step 7.3

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

$- 5 \frac{1}{\frac{lim_{x \to \infty} \sqrt{1 - \frac{5}{x^{2}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 7.4

Move the limit under the radical sign.

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} 1 - \frac{5}{x^{2}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 7.5

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

$- 5 \frac{1}{\frac{\sqrt{lim_{x \to \infty} 1 - lim_{x \to \infty} \frac{5}{x^{2}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 7.6

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

$- 5 \frac{1}{\frac{\sqrt{1 - lim_{x \to \infty} \frac{5}{x^{2}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 7.7

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$- 5 \frac{1}{\frac{\sqrt{1 - 5 lim_{x \to \infty} \frac{1}{x^{2}}}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 8

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

$- 5 \frac{1}{\frac{\sqrt{1 - 5 \cdot 0}}{lim_{x \to \infty} 1} + lim_{x \to \infty} 1}$

Step 9

Evaluate the limit.

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Step 9.1

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

$- 5 \frac{1}{\frac{\sqrt{1 - 5 \cdot 0}}{1} + lim_{x \to \infty} 1}$

Step 9.2

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

$- 5 \frac{1}{\frac{\sqrt{1 - 5 \cdot 0}}{1} + 1}$

Step 9.3

Simplify the answer.

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Step 9.3.1

Divide $\sqrt{1 - 5 \cdot 0}$ by $1$ .

$- 5 \frac{1}{\sqrt{1 - 5 \cdot 0} + 1}$

Step 9.3.2

Simplify the denominator.

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Step 9.3.2.1

Multiply $- 5$ by $0$ .

$- 5 \frac{1}{\sqrt{1 + 0} + 1}$

Step 9.3.2.2

Add $1$ and $0$ .

$- 5 \frac{1}{\sqrt{1} + 1}$

Step 9.3.2.3

Any root of $1$ is $1$ .

$- 5 \frac{1}{1 + 1}$

Step 9.3.2.4

Add $1$ and $1$ .

$- 5 (\frac{1}{2})$

Step 9.3.3

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{- 5}{2}$

Step 9.3.4

Move the negative in front of the fraction.

$- \frac{5}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{5}{2}$

Decimal Form:

$- 2.5$