Calculus Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Expand the numerator using the FOIL method.

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

Step 2.2

Simplify.

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

Step 3

Evaluate the limit.

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Step 3.1

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

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Step 3.1.1

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

$lim_{x \to \infty} \frac{3 x}{x + \sqrt{x \cdot x - 3 x}}$

Step 3.1.2

Factor $x$ out of $- 3 x$ .

$lim_{x \to \infty} \frac{3 x}{x + \sqrt{x \cdot x + x \cdot - 3}}$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $x$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x$ .

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

Step 6

Cancel the common factors.

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Evaluate the limit.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

Move the limit under the radical sign.

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

Step 8

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

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

Step 9

Evaluate the limit.

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Step 9.1

Simplify each term.

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Step 9.1.1

Cancel the common factor of $x$ .

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Step 9.1.1.1

Cancel the common factor.

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

Step 9.1.1.2

Rewrite the expression.

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

Step 9.1.2

Move the negative in front of the fraction.

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

Step 9.2

Cancel the common factor of $x$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Rewrite the expression.

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

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

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

Step 10

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

$3 \frac{1}{1 + \sqrt{\frac{1 - 3 \cdot 0}{lim_{x \to \infty} 1}}}$

Step 11

Evaluate the limit.

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Step 11.1

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

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

Step 11.2

Simplify the answer.

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Step 11.2.1

Divide $1 - 3 \cdot 0$ by $1$ .

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

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

Multiply $- 3$ by $0$ .

$3 \frac{1}{1 + \sqrt{1 + 0}}$

Step 11.2.2.2

Add $1$ and $0$ .

$3 \frac{1}{1 + \sqrt{1}}$

Step 11.2.2.3

Any root of $1$ is $1$ .

$3 \frac{1}{1 + 1}$

Step 11.2.2.4

Add $1$ and $1$ .

$3 (\frac{1}{2})$

Step 11.2.3

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2}$