Calculus Examples

$lim_{x \to \infty} \frac{\sqrt{5 x + 4} - \sqrt{3 x + 9}}{4 x}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{5 x + 4} - \sqrt{3 x + 9}}{x}$

Step 1.2

Factor $3$ out of $3 x + 9$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{5 x + 4} - \sqrt{3 (x) + 9}}{x}$

Step 1.2.2

Factor $3$ out of $9$ .

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{5 x + 4} - \sqrt{3 x + 3 \cdot 3}}{x}$

Step 1.2.3

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

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{5 x + 4} - \sqrt{3 (x + 3)}}{x}$

Step 2

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

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{5 x}{x^{2}} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{\frac{x}{x}}$

Step 3

Evaluate the limit.

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Step 3.1

Cancel the common factor of $x$ .

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{5 x}{x^{2}} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{1}$

Step 3.2

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

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Step 3.2.1

Factor $x$ out of $5 x$ .

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 5}{x^{2}} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{1}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 5}{x \cdot x} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{1}$

Step 3.2.2.2

Cancel the common factor.

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{x \cdot 5}{x \cdot x} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{1}$

Step 3.2.2.3

Rewrite the expression.

$\frac{1}{4} lim_{x \to \infty} \frac{\sqrt{\frac{5}{x} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{1}$

Step 3.3

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

$\frac{1}{4} \cdot \frac{lim_{x \to \infty} \sqrt{\frac{5}{x} + \frac{4}{x^{2}}} - \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 3.4

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

$\frac{1}{4} \cdot \frac{lim_{x \to \infty} \sqrt{\frac{5}{x} + \frac{4}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 3.5

Move the limit under the radical sign.

$\frac{1}{4} \cdot \frac{\sqrt{lim_{x \to \infty} \frac{5}{x} + \frac{4}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 3.6

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

$\frac{1}{4} \cdot \frac{\sqrt{lim_{x \to \infty} \frac{5}{x} + lim_{x \to \infty} \frac{4}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 3.7

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

$\frac{1}{4} \cdot \frac{\sqrt{5 lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{4}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 4

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + lim_{x \to \infty} \frac{4}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 lim_{x \to \infty} \frac{1}{x^{2}}} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 6

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - lim_{x \to \infty} \sqrt{\frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 7

Evaluate the limit.

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Step 7.1

Move the limit under the radical sign.

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{lim_{x \to \infty} \frac{3 (x + 3)}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 7.2

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{x + 3}{x^{2}}}}{lim_{x \to \infty} 1}$

Step 8

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{x}{x^{2}} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9

Evaluate the limit.

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Step 9.1

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

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Step 9.1.1

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{x^{1}}{x^{2}} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.1.2

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x^{2}} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.1.3

Cancel the common factors.

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Step 9.1.3.1

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x \cdot x} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.1.3.2

Cancel the common factor.

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{x \cdot 1}{x \cdot x} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.1.3.3

Rewrite the expression.

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{1}{x} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.2

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

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Step 9.2.1

Cancel the common factor.

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{1}{x} + \frac{3}{x^{2}}}{\frac{x^{2}}{x^{2}}}}}{lim_{x \to \infty} 1}$

Step 9.2.2

Rewrite the expression.

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 lim_{x \to \infty} \frac{\frac{1}{x} + \frac{3}{x^{2}}}{1}}}{lim_{x \to \infty} 1}$

Step 9.3

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{lim_{x \to \infty} \frac{1}{x} + \frac{3}{x^{2}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 9.4

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 10

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{0 + lim_{x \to \infty} \frac{3}{x^{2}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 11

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{0 + 3 lim_{x \to \infty} \frac{1}{x^{2}}}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 12

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{0 + 3 \cdot 0}{lim_{x \to \infty} 1}}}{lim_{x \to \infty} 1}$

Step 13

Evaluate the limit.

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Step 13.1

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{0 + 3 \cdot 0}{1}}}{lim_{x \to \infty} 1}$

Step 13.2

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 \frac{0 + 3 \cdot 0}{1}}}{1}$

Step 13.3

Simplify the answer.

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Step 13.3.1

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

$\frac{1}{4} \cdot \frac{\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 (0 + 3 \cdot 0)}}{1}$

Step 13.3.2

Divide $\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 (0 + 3 \cdot 0)}$ by $1$ .

$\frac{1}{4} (\sqrt{5 \cdot 0 + 4 \cdot 0} - \sqrt{3 (0 + 3 \cdot 0)})$

Step 13.3.3

Simplify each term.

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Step 13.3.3.1

Multiply $5$ by $0$ .

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

Step 13.3.3.2

Multiply $4$ by $0$ .

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

Step 13.3.3.3

Add $0$ and $0$ .

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

Step 13.3.3.4

Rewrite $0$ as $0^{2}$ .

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

Step 13.3.3.5

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

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

Step 13.3.3.6

Multiply $3$ by $0$ .

$\frac{1}{4} (0 - \sqrt{3 (0 + 0)})$

Step 13.3.3.7

Add $0$ and $0$ .

$\frac{1}{4} (0 - \sqrt{3 \cdot 0})$

Step 13.3.3.8

Multiply $3$ by $0$ .

$\frac{1}{4} (0 - \sqrt{0})$

Step 13.3.3.9

Rewrite $0$ as $0^{2}$ .

$\frac{1}{4} (0 - \sqrt{0^{2}})$

Step 13.3.3.10

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

$\frac{1}{4} (0 - 0)$

Step 13.3.3.11

Multiply $- 1$ by $0$ .

$\frac{1}{4} (0 + 0)$

Step 13.3.4

Add $0$ and $0$ .

$\frac{1}{4} \cdot 0$

Step 13.3.5

Multiply $\frac{1}{4}$ by $0$ .

$0$