Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Evaluate the limit of $x^{3}$ which is constant as $h$ approaches $0$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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 $a = x + 0$ and $b = x$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

Add $x$ and $0$ .

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

Step 3.1.2.2

Subtract $x$ from $x$ .

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

Step 3.1.2.3

Add $x$ and $0$ .

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

Step 3.1.2.4

Add $x$ and $0$ .

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

Step 3.1.2.5

Multiply $x$ by $x$ .

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

Step 3.1.2.6

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

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

Step 3.1.2.7

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

$\frac{0 \cdot 3 x^{2}}{h}$

Step 3.1.2.8

Combine exponents.

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Step 3.1.2.8.1

Multiply $0$ by $3$ .

$\frac{0 x^{2}}{h}$

Step 3.1.2.8.2

Multiply $0$ by $x^{2}$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$