Calculus Examples

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

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 $0$ .

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

Step 1.2

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

Evaluate the limit of $2 a^{2}$ which is constant as $h$ approaches $0$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $2$ out of $2 {(a + 0)}^{2} - 2 a^{2}$ .

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Step 3.1.1.1

Factor $2$ out of $2 {(a + 0)}^{2}$ .

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

Step 3.1.1.2

Factor $2$ out of $- 2 a^{2}$ .

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

Step 3.1.1.3

Factor $2$ out of $2 ({(a + 0)}^{2}) + 2 (- a^{2})$ .

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

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a + 0$ and $b = a$ .

$\frac{2 ((a + 0 + a) (a + 0 - a))}{h}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Add $a$ and $0$ .

$\frac{2 ((a + a) (a + 0 - a))}{h}$

Step 3.1.3.2

Add $a$ and $a$ .

$\frac{2 (2 a (a + 0 - a))}{h}$

Step 3.1.3.3

Add $a$ and $0$ .

$\frac{2 (2 a (a - a))}{h}$

Step 3.1.3.4

Subtract $a$ from $a$ .

$\frac{2 (2 a \cdot 0)}{h}$

Step 3.1.3.5

Combine exponents.

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Step 3.1.3.5.1

Multiply $0$ by $2$ .

$\frac{2 (0 a)}{h}$

Step 3.1.3.5.2

Multiply $0$ by $a$ .

$\frac{2 \cdot 0}{h}$

Step 3.2

Multiply $2$ by $0$ .

$\frac{0}{h}$

Step 3.3

Divide $0$ by $h$ .

$0$