Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Add $0$ and $h$ .

$\frac{h^{3} - 0^{3}}{h}$

Step 7.1.2

Raising $0$ to any positive power yields $0$ .

$\frac{h^{3} - 0}{h}$

Step 7.1.3

Multiply $- 1$ by $0$ .

$\frac{h^{3} + 0}{h}$

Step 7.1.4

Add $h^{3}$ and $0$ .

$\frac{h^{3}}{h}$

Step 7.2

Cancel the common factor of $h^{3}$ and $h$ .

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Step 7.2.1

Factor $h$ out of $h^{3}$ .

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

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

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

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

Step 7.2.2.2

Factor $h$ out of $h^{1}$ .

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

Step 7.2.2.3

Cancel the common factor.

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

Step 7.2.2.4

Rewrite the expression.

$\frac{h^{2}}{1}$

Step 7.2.2.5

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

$h^{2}$