Calculus Examples

$\frac{lim_{h \to 0} \sqrt{64 + h} - 8}{h}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{h \to 0} \sqrt{64 + h} - lim_{h \to 0} 8}{h}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{h \to 0} 64 + h} - lim_{h \to 0} 8}{h}$

Step 1.3

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

$\frac{\sqrt{lim_{h \to 0} 64 + lim_{h \to 0} h} - lim_{h \to 0} 8}{h}$

Step 1.4

Evaluate the limit of $64$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{64 + lim_{h \to 0} h} - lim_{h \to 0} 8}{h}$

Step 1.5

Evaluate the limit of $8$ which is constant as $h$ approaches $0$ .

$\frac{\sqrt{64 + lim_{h \to 0} h} - 1 \cdot 8}{h}$

Step 2

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

$\frac{\sqrt{64 + 0} - 1 \cdot 8}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $64$ and $0$ .

$\frac{\sqrt{64} - 1 \cdot 8}{h}$

Step 3.1.2

Rewrite $64$ as $8^{2}$ .

$\frac{\sqrt{8^{2}} - 1 \cdot 8}{h}$

Step 3.1.3

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

$\frac{8 - 1 \cdot 8}{h}$

Step 3.1.4

Multiply $- 1$ by $8$ .

$\frac{8 - 8}{h}$

Step 3.1.5

Subtract $8$ from $8$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$