Calculus Examples

$\frac{lim_{x \to 0} {(1 + h)}^{4} - 1}{h}$

Step 1

Evaluate the limit of ${(1 + h)}^{4} - 1$ which is constant as $x$ approaches $0$ .

$\frac{{(1 + h)}^{4} - 1}{h}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite ${(1 + h)}^{4}$ as ${({(1 + h)}^{2})}^{2}$ .

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

Step 2.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.3

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

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

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

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

Step 2.1.4.2

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

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Step 2.1.4.2.1

Apply the distributive property.

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

Step 2.1.4.2.2

Apply the distributive property.

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

Step 2.1.4.2.3

Apply the distributive property.

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

Step 2.1.4.3

Simplify and combine like terms.

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Step 2.1.4.3.1

Simplify each term.

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Step 2.1.4.3.1.1

Multiply $1$ by $1$ .

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

Step 2.1.4.3.1.2

Multiply $h$ by $1$ .

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

Step 2.1.4.3.1.3

Multiply $h$ by $1$ .

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

Step 2.1.4.3.1.4

Multiply $h$ by $h$ .

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

Step 2.1.4.3.2

Add $h$ and $h$ .

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

Step 2.1.4.4

Add $1$ and $1$ .

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

Step 2.1.4.5

Reorder terms.

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

Step 2.1.4.6

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.4.7

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

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

Step 2.1.4.8

Simplify.

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Step 2.1.4.8.1

Add $1$ and $1$ .

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

Step 2.1.4.8.2

Subtract $1$ from $1$ .

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

Step 2.1.4.8.3

Add $h$ and $0$ .

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

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

Step 2.2

Cancel the common factor of $h$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $(h^{2} + 2 h + 2) (h + 2)$ by $1$ .

$(h^{2} + 2 h + 2) (h + 2)$

Step 2.3

Expand $(h^{2} + 2 h + 2) (h + 2)$ by multiplying each term in the first expression by each term in the second expression.

$h^{2} h + h^{2} \cdot 2 + 2 h \cdot h + 2 h \cdot 2 + 2 h + 2 \cdot 2$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $h^{2}$ by $h$ by adding the exponents.

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Step 2.4.1.1

Multiply $h^{2}$ by $h$ .

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Step 2.4.1.1.1

Raise $h$ to the power of $1$ .

$h^{2} h^{1} + h^{2} \cdot 2 + 2 h \cdot h + 2 h \cdot 2 + 2 h + 2 \cdot 2$

Step 2.4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$h^{2 + 1} + h^{2} \cdot 2 + 2 h \cdot h + 2 h \cdot 2 + 2 h + 2 \cdot 2$

Step 2.4.1.2

Add $2$ and $1$ .

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

Step 2.4.2

Move $2$ to the left of $h^{2}$ .

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

Step 2.4.3

Multiply $h$ by $h$ by adding the exponents.

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Step 2.4.3.1

Move $h$ .

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

Step 2.4.3.2

Multiply $h$ by $h$ .

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

Step 2.4.4

Multiply $2$ by $2$ .

$h^{3} + 2 h^{2} + 2 h^{2} + 4 h + 2 h + 2 \cdot 2$

Step 2.4.5

Multiply $2$ by $2$ .

$h^{3} + 2 h^{2} + 2 h^{2} + 4 h + 2 h + 4$

Step 2.5

Add $2 h^{2}$ and $2 h^{2}$ .

$h^{3} + 4 h^{2} + 4 h + 2 h + 4$

Step 2.6

Add $4 h$ and $2 h$ .

$h^{3} + 4 h^{2} + 6 h + 4$