Calculus Examples

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

Step 1

Evaluate the limit of $\frac{{(1 + h)}^{3} - 1}{h}$ which is constant as $x$ approaches $0$ .

$\frac{{(1 + h)}^{3} - 1}{h}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $1$ as $1^{3}$ .

$\frac{{(1 + h)}^{3} - 1^{3}}{h}$

Step 2.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1 + h$ and $b = 1$ .

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

Step 2.1.3

Simplify.

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Step 2.1.3.1

Subtract $1$ from $1$ .

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

Step 2.1.3.2

Add $h$ and $0$ .

$\frac{h ({(1 + h)}^{2} + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.3

Rewrite ${(1 + h)}^{2}$ as $(1 + h) (1 + h)$ .

$\frac{h ((1 + h) (1 + h) + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.4

Expand $(1 + h) (1 + h)$ using the FOIL Method.

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Step 2.1.3.4.1

Apply the distributive property.

$\frac{h (1 (1 + h) + h (1 + h) + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.4.2

Apply the distributive property.

$\frac{h (1 \cdot 1 + 1 h + h (1 + h) + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.4.3

Apply the distributive property.

$\frac{h (1 \cdot 1 + 1 h + h \cdot 1 + h \cdot h + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.5

Simplify and combine like terms.

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Step 2.1.3.5.1

Simplify each term.

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Step 2.1.3.5.1.1

Multiply $1$ by $1$ .

$\frac{h (1 + 1 h + h \cdot 1 + h \cdot h + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.5.1.2

Multiply $h$ by $1$ .

$\frac{h (1 + h + h \cdot 1 + h \cdot h + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.5.1.3

Multiply $h$ by $1$ .

$\frac{h (1 + h + h + h \cdot h + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.5.1.4

Multiply $h$ by $h$ .

$\frac{h (1 + h + h + h^{2} + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.5.2

Add $h$ and $h$ .

$\frac{h (1 + 2 h + h^{2} + (1 + h) \cdot 1 + 1^{2})}{h}$

Step 2.1.3.6

Multiply $1 + h$ by $1$ .

$\frac{h (1 + 2 h + h^{2} + 1 + h + 1^{2})}{h}$

Step 2.1.3.7

One to any power is one.

$\frac{h (1 + 2 h + h^{2} + 1 + h + 1)}{h}$

Step 2.1.3.8

Add $1$ and $1$ .

$\frac{h (2 h + h^{2} + 2 + h + 1)}{h}$

Step 2.1.3.9

Add $2 h$ and $h$ .

$\frac{h (3 h + h^{2} + 2 + 1)}{h}$

Step 2.1.3.10

Add $2$ and $1$ .

$\frac{h (3 h + h^{2} + 3)}{h}$

Step 2.1.3.11

Reorder terms.

$\frac{h (h^{2} + 3 h + 3)}{h}$

Step 2.2

Cancel the common factor of $h$ .

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Step 2.2.1

Cancel the common factor.

$\frac{h (h^{2} + 3 h + 3)}{h}$

Step 2.2.2

Divide $h^{2} + 3 h + 3$ by $1$ .

$h^{2} + 3 h + 3$