Calculus Examples

$\frac{lim_{h \to 0} {(3 + h)}^{2} - 9}{h}$

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 $0$ .

$\frac{lim_{h \to 0} {(3 + h)}^{2} - lim_{h \to 0} 9}{h}$

Step 1.2

Move the exponent $2$ from ${(3 + h)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{h \to 0} 3 + h)}^{2} - lim_{h \to 0} 9}{h}$

Step 1.3

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

$\frac{{(lim_{h \to 0} 3 + lim_{h \to 0} h)}^{2} - lim_{h \to 0} 9}{h}$

Step 1.4

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$\frac{{(3 + lim_{h \to 0} h)}^{2} - lim_{h \to 0} 9}{h}$

Step 1.5

Evaluate the limit of $9$ which is constant as $h$ approaches $0$ .

$\frac{{(3 + lim_{h \to 0} h)}^{2} - 1 \cdot 9}{h}$

Step 2

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

$\frac{{(3 + 0)}^{2} - 1 \cdot 9}{h}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $3$ and $0$ .

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

Step 3.1.2

Raise $3$ to the power of $2$ .

$\frac{9 - 1 \cdot 9}{h}$

Step 3.1.3

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{h}$

Step 3.1.4

Subtract $9$ from $9$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$