Calculus Examples

$lim_{h \to 0} \frac{(\sqrt{7 + h} - \sqrt{7}) (\sqrt{7 + h} + \sqrt{7})}{h (\sqrt{7 + h} + 7)}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} (\sqrt{7 + h} - \sqrt{7}) (\sqrt{7 + h} + \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 2

Split the limit using the Product of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} \sqrt{7 + h} - \sqrt{7} \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(lim_{h \to 0} \sqrt{7 + h} - lim_{h \to 0} \sqrt{7}) \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 4

Move the limit under the radical sign.

$\frac{(\sqrt{lim_{h \to 0} 7 + h} - lim_{h \to 0} \sqrt{7}) \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt{lim_{h \to 0} 7 + lim_{h \to 0} h} - lim_{h \to 0} \sqrt{7}) \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 6

Evaluate the limit of $7$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - lim_{h \to 0} \sqrt{7}) \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 7

Evaluate the limit of $\sqrt{7}$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot lim_{h \to 0} \sqrt{7 + h} + \sqrt{7}}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 8

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot (lim_{h \to 0} \sqrt{7 + h} + lim_{h \to 0} \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 9

Move the limit under the radical sign.

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot (\sqrt{lim_{h \to 0} 7 + h} + lim_{h \to 0} \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 10

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot (\sqrt{lim_{h \to 0} 7 + lim_{h \to 0} h} + lim_{h \to 0} \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 11

Evaluate the limit of $7$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot (\sqrt{7 + lim_{h \to 0} h} + lim_{h \to 0} \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 12

Evaluate the limit of $\sqrt{7}$ which is constant as $h$ approaches $0$ .

$\frac{(\sqrt{7 + lim_{h \to 0} h} - \sqrt{7}) \cdot (\sqrt{7 + lim_{h \to 0} h} + \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 13

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 13.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{(\sqrt{7 + 0} - \sqrt{7}) \cdot (\sqrt{7 + lim_{h \to 0} h} + \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 13.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{(\sqrt{7 + 0} - \sqrt{7}) \cdot (\sqrt{7 + 0} + \sqrt{7})}{lim_{h \to 0} h (\sqrt{7 + h} + 7)}$

Step 13.3

Evaluate the limit of $h (\sqrt{7 + h} + 7)$ by plugging in $0$ for $h$ .

$\frac{(\sqrt{7 + 0} - \sqrt{7}) \cdot (\sqrt{7 + 0} + \sqrt{7})}{h (\sqrt{7 + 0} + 7)}$

Step 13.4

Add $7$ and $0$ .

$\frac{(\sqrt{7 + 0} - \sqrt{7}) \cdot (\sqrt{7 + 0} + \sqrt{7})}{h (\sqrt{7} + 7)}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

Add $7$ and $0$ .

$\frac{(\sqrt{7} - \sqrt{7}) (\sqrt{7 + 0} + \sqrt{7})}{h (\sqrt{7} + 7)}$

Step 14.1.2

Subtract $\sqrt{7}$ from $\sqrt{7}$ .

$\frac{0 (\sqrt{7 + 0} + \sqrt{7})}{h (\sqrt{7} + 7)}$

Step 14.1.3

Add $7$ and $0$ .

$\frac{0 (\sqrt{7} + \sqrt{7})}{h (\sqrt{7} + 7)}$

Step 14.1.4

Add $\sqrt{7}$ and $\sqrt{7}$ .

$\frac{0 \cdot 2 \sqrt{7}}{h (\sqrt{7} + 7)}$

Step 14.1.5

Combine exponents.

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Step 14.1.5.1

Multiply $0$ by $2$ .

$\frac{0 \sqrt{7}}{h (\sqrt{7} + 7)}$

Step 14.1.5.2

Multiply $0$ by $\sqrt{7}$ .

$\frac{0}{h (\sqrt{7} + 7)}$

Step 14.2

Divide $0$ by $h (\sqrt{7} + 7)$ .

$0$