Calculus Examples

$f (x) = \frac{1}{4 - x^{2}}$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{1}{4 - {(x + h)}^{2}}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.1.2.1.2

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

$f (x + h) = \frac{1}{(2 + x + h) (2 - (x + h))}$

Step 2.1.2.1.3

Apply the distributive property.

$f (x + h) = \frac{1}{(2 + x + h) (2 - x - h)}$

Step 2.1.2.2

The final answer is $\frac{1}{(2 + x + h) (2 - x - h)}$ .

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

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{1}{(2 + x + h) (2 - x - h)}$

$f (x) = \frac{1}{(2 + x) (2 - x)}$

$f (x + h) = \frac{1}{(2 + x + h) (2 - x - h)}$

$f (x) = \frac{1}{(2 + x) (2 - x)}$

Step 3

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{1}{(2 + x + h) (2 - x - h)} - (\frac{1}{(2 + x) (2 - x)})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{(2 + x + h) (2 - x - h)}$ as a fraction with a common denominator, multiply by $\frac{(2 + x) (2 - x)}{(2 + x) (2 - x)}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1}{(2 + x + h) (2 - x - h)} \cdot \frac{(2 + x) (2 - x)}{(2 + x) (2 - x)} - \frac{1}{(2 + x) (2 - x)}}{h}$

Step 4.1.2

To write $- \frac{1}{(2 + x) (2 - x)}$ as a fraction with a common denominator, multiply by $\frac{(2 + x + h) (2 - x - h)}{(2 + x + h) (2 - x - h)}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{1}{(2 + x + h) (2 - x - h)} \cdot \frac{(2 + x) (2 - x)}{(2 + x) (2 - x)} - \frac{1}{(2 + x) (2 - x)} \cdot \frac{(2 + x + h) (2 - x - h)}{(2 + x + h) (2 - x - h)}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(2 + x + h) (2 - x - h) (2 + x) (2 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{1}{(2 + x + h) (2 - x - h)}$ by $\frac{(2 + x) (2 - x)}{(2 + x) (2 - x)}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{(2 + x) (2 - x)}{(2 + x + h) (2 - x - h) ((2 + x) (2 - x))} - \frac{1}{(2 + x) (2 - x)} \cdot \frac{(2 + x + h) (2 - x - h)}{(2 + x + h) (2 - x - h)}}{h}$

Step 4.1.3.2

Multiply $\frac{1}{(2 + x) (2 - x)}$ by $\frac{(2 + x + h) (2 - x - h)}{(2 + x + h) (2 - x - h)}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{(2 + x) (2 - x)}{(2 + x + h) (2 - x - h) ((2 + x) (2 - x))} - \frac{(2 + x + h) (2 - x - h)}{(2 + x) (2 - x) ((2 + x + h) (2 - x - h))}}{h}$

Step 4.1.3.3

Reorder the factors of $(2 + x + h) (2 - x - h) ((2 + x) (2 - x))$ .

$f' (x) = lim_{h \to 0} \frac{\frac{(2 + x) (2 - x)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)} - \frac{(2 + x + h) (2 - x - h)}{(2 + x) (2 - x) ((2 + x + h) (2 - x - h))}}{h}$

Step 4.1.3.4

Reorder the factors of $(2 + x) (2 - x) ((2 + x + h) (2 - x - h))$ .

$f' (x) = lim_{h \to 0} \frac{\frac{(2 + x) (2 - x)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)} - \frac{(2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{(2 + x) (2 - x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5

Simplify the numerator.

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Step 4.1.5.1

Expand $(2 + x) (2 - x)$ using the FOIL Method.

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Step 4.1.5.1.1

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 (2 - x) + x (2 - x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.1.2

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 \cdot 2 + 2 (- x) + x (2 - x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.1.3

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{2 \cdot 2 + 2 (- x) + x \cdot 2 + x (- x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2

Simplify and combine like terms.

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Step 4.1.5.2.1

Simplify each term.

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Step 4.1.5.2.1.1

Multiply $2$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 + 2 (- x) + x \cdot 2 + x (- x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.1.2

Multiply $- 1$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - 2 x + x \cdot 2 + x (- x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.1.3

Move $2$ to the left of $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - 2 x + 2 \cdot x + x (- x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.1.4

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - 2 x + 2 x - x \cdot x - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.2.1.5.1

Move $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - 2 x + 2 x - (x \cdot x) - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.1.5.2

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - 2 x + 2 x - x^{2} - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.2

Add $- 2 x$ and $2 x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 + 0 - x^{2} - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.2.3

Add $4$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - (2 + x + h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.3

Apply the distributive property.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} + (- 1 \cdot 2 - x - h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.4

Multiply $- 1$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} + (- 2 - x - h) (2 - x - h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.5

Expand $(- 2 - x - h) (2 - x - h)$ by multiplying each term in the first expression by each term in the second expression.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 2 \cdot 2 - 2 (- x) - 2 (- h) - x \cdot 2 - x (- x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6

Simplify each term.

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Step 4.1.5.6.1

Multiply $- 2$ by $2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 - 2 (- x) - 2 (- h) - x \cdot 2 - x (- x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.2

Multiply $- 1$ by $- 2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x - 2 (- h) - x \cdot 2 - x (- x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.3

Multiply $- 1$ by $- 2$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - x \cdot 2 - x (- x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.4

Multiply $2$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x - x (- x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.5

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x - 1 \cdot (- 1 x \cdot x) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.6.6.1

Move $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x - 1 \cdot (- 1 (x \cdot x)) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.6.2

Multiply $x$ by $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x - 1 \cdot (- 1 x^{2}) - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.7

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + 1 x^{2} - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.8

Multiply $x^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} - x (- h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.9

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} - 1 \cdot (- 1 x h) - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.10

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + 1 x h - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.11

Multiply $x$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - h \cdot 2 - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.12

Multiply $2$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h - h (- x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.13

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h - 1 \cdot (- 1 h x) - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.14

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + 1 h x - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.15

Multiply $h$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x - h (- h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.16

Rewrite using the commutative property of multiplication.

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x - 1 \cdot (- 1 h \cdot h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.17

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

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Step 4.1.5.6.17.1

Move $h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x - 1 \cdot (- 1 (h \cdot h))}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.17.2

Multiply $h$ by $h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x - 1 \cdot (- 1 h^{2})}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.18

Multiply $- 1$ by $- 1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x + 1 h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.6.19

Multiply $h^{2}$ by $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.7

Combine the opposite terms in $- 4 + 2 x + 2 h - 2 x + x^{2} + x h - 2 h + h x + h^{2}$ .

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Step 4.1.5.7.1

Subtract $2 x$ from $2 x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 h + 0 + x^{2} + x h - 2 h + h x + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.7.2

Add $- 4 + 2 h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + 2 h + x^{2} + x h - 2 h + h x + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.7.3

Subtract $2 h$ from $2 h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + x^{2} + x h + 0 + h x + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.7.4

Add $- 4 + x^{2} + x h$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + x^{2} + x h + h x + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.8

Add $x h$ and $h x$ .

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Step 4.1.5.8.1

Reorder $h$ and $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + x^{2} + x h + x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.8.2

Add $x h$ and $x h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{4 - x^{2} - 4 + x^{2} + 2 x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.9

Subtract $4$ from $4$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- x^{2} + 0 + x^{2} + 2 x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.10

Add $- x^{2}$ and $0$ .

$f' (x) = lim_{h \to 0} \frac{\frac{- x^{2} + x^{2} + 2 x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.11

Add $- x^{2}$ and $x^{2}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{0 + 2 x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.12

Add $0$ and $2 x h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{2 x h + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.13

Factor $h$ out of $2 x h + h^{2}$ .

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Step 4.1.5.13.1

Factor $h$ out of $2 x h$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (2 x) + h^{2}}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.13.2

Factor $h$ out of $h^{2}$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (2 x) + h (h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.1.5.13.3

Factor $h$ out of $h (2 x) + h (h)$ .

$f' (x) = lim_{h \to 0} \frac{\frac{h (2 x + h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$f' (x) = lim_{h \to 0} \frac{h (2 x + h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$f' (x) = lim_{h \to 0} \frac{h (2 x + h)}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$f' (x) = lim_{h \to 0} \frac{2 x + h}{(2 + x) (2 + x + h) (2 - x) (2 - x - h)}$

Step 5

Move the term $\frac{1}{(2 + x) (2 - x)}$ outside of the limit because it is constant with respect to $h$ .

$\frac{1}{(2 + x) (2 - x)} lim_{h \to 0} \frac{2 x + h}{(2 + x + h) (2 - x - h)}$

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

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

Step 12

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

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

Step 13

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

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

Step 14

Evaluate the limit of $2$ which is constant as $h$ approaches $0$ .

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

Step 15

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

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

Step 16

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 16.1

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

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

Step 16.2

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

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

Step 16.3

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

$\frac{1}{(2 + x) (2 - x)} \cdot \frac{2 x + 0}{(2 + x + 0) \cdot (2 - x + 0)}$

Step 17

Simplify the answer.

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Step 17.1

Add $2 x$ and $0$ .

$\frac{1}{(2 + x) (2 - x)} \cdot \frac{2 x}{(2 + x + 0) \cdot (2 - x + 0)}$

Step 17.2

Simplify the denominator.

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Step 17.2.1

Add $2 + x$ and $0$ .

$\frac{1}{(2 + x) (2 - x)} \cdot \frac{2 x}{(2 + x) (2 - x + 0)}$

Step 17.2.2

Add $2 - x$ and $0$ .

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

Step 17.3

Multiply $\frac{1}{(2 + x) (2 - x)} \cdot \frac{2 x}{(2 + x) (2 - x)}$ .

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Step 17.3.1

Multiply $\frac{1}{(2 + x) (2 - x)}$ by $\frac{2 x}{(2 + x) (2 - x)}$ .

$\frac{2 x}{(2 + x) (2 - x) ((2 + x) (2 - x))}$

Step 17.3.2

Raise $2 + x$ to the power of $1$ .

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

Step 17.3.3

Raise $2 + x$ to the power of $1$ .

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

Step 17.3.4

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

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

Step 17.3.5

Add $1$ and $1$ .

$\frac{2 x}{{(2 + x)}^{2} (2 - x) (2 - x)}$

Step 17.3.6

Raise $2 - x$ to the power of $1$ .

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

Step 17.3.7

Raise $2 - x$ to the power of $1$ .

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

Step 17.3.8

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

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

Step 17.3.9

Add $1$ and $1$ .

$\frac{2 x}{{(2 + x)}^{2} {(2 - x)}^{2}}$

Step 18