Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

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

Step 2

Move the limit under the radical sign.

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

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

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

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $5$ .

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

Step 5

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

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

Step 5.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 1$ by $5$ .

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

Step 6.1.2

Subtract $5$ from $5$ .

$\frac{\sqrt{0}}{5}$

Step 6.1.3

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

$\frac{\sqrt{0^{2}}}{5}$

Step 6.1.4

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

$\frac{0}{5}$

Step 6.2

Divide $0$ by $5$ .

$0$