Calculus Examples

$\frac{lim_{h \to 2} (\sqrt{9 + h}) - 3}{h}$

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 $2$ .

$\frac{lim_{h \to 2} \sqrt{9 + h} - lim_{h \to 2} 3}{h}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{h \to 2} 9 + h} - lim_{h \to 2} 3}{h}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $2$ .

$\frac{\sqrt{lim_{h \to 2} 9 + lim_{h \to 2} h} - lim_{h \to 2} 3}{h}$

Step 1.4

Evaluate the limit of $9$ which is constant as $h$ approaches $2$ .

$\frac{\sqrt{9 + lim_{h \to 2} h} - lim_{h \to 2} 3}{h}$

Step 1.5

Evaluate the limit of $3$ which is constant as $h$ approaches $2$ .

$\frac{\sqrt{9 + lim_{h \to 2} h} - 1 \cdot 3}{h}$

Step 2

Evaluate the limit of $h$ by plugging in $2$ for $h$ .

$\frac{\sqrt{9 + 2} - 1 \cdot 3}{h}$

Step 3

Simplify the numerator.

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Step 3.1

Add $9$ and $2$ .

$\frac{\sqrt{11} - 1 \cdot 3}{h}$

Step 3.2

Multiply $- 1$ by $3$ .

$\frac{\sqrt{11} - 3}{h}$