Calculus Examples

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

Step 1

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

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

Step 2

Evaluate the limit.

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Step 2.1

Cancel the common factor of $x$ .

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Step 2.1.1

Cancel the common factor.

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

Step 2.1.2

Rewrite the expression.

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Move the limit under the radical sign.

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

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

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

Step 4

Evaluate the limit.

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Step 4.1

Move the limit under the radical sign.

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

Step 4.2

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

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

Step 5

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

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

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 $\infty$ .

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

Step 6.2

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

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

Step 6.3

Move the limit under the radical sign.

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

Step 6.4

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $5$ by $0$ .

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

Step 8.1.2

Add $1$ and $0$ .

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

Step 8.1.3

Any root of $1$ is $1$ .

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

Step 8.1.4

Multiply $5$ by $0$ .

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

Step 8.1.5

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

$\frac{1 - \sqrt{0^{2}}}{\sqrt{1} - \sqrt{5 \cdot 0}}$

Step 8.1.6

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

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

Step 8.1.7

Multiply $- 1$ by $0$ .

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

Step 8.1.8

Add $1$ and $0$ .

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

Step 8.2

Simplify the denominator.

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Step 8.2.1

Any root of $1$ is $1$ .

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

Step 8.2.2

Multiply $5$ by $0$ .

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

Step 8.2.3

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

$\frac{1}{1 - \sqrt{0^{2}}}$

Step 8.2.4

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

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

Step 8.2.5

Multiply $- 1$ by $0$ .

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

Step 8.2.6

Add $1$ and $0$ .

$\frac{1}{1}$

Step 8.3

Divide $1$ by $1$ .

$1$