Calculus Examples

$lim_{x \to 0} \frac{2 {(x + h)}^{3} - 2 x^{3}}{h}$

Step 1

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

$\frac{1}{h} lim_{x \to 0} 2 {(x + h)}^{3} - 2 x^{3}$

Step 2

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

$\frac{1}{h} (lim_{x \to 0} 2 {(x + h)}^{3} - lim_{x \to 0} 2 x^{3})$

Step 3

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

$\frac{1}{h} (2 lim_{x \to 0} {(x + h)}^{3} - lim_{x \to 0} 2 x^{3})$

Step 4

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

$\frac{1}{h} (2 {(lim_{x \to 0} x + h)}^{3} - lim_{x \to 0} 2 x^{3})$

Step 5

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

$\frac{1}{h} (2 {(lim_{x \to 0} x + lim_{x \to 0} h)}^{3} - lim_{x \to 0} 2 x^{3})$

Step 6

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

$\frac{1}{h} (2 {(lim_{x \to 0} x + h)}^{3} - lim_{x \to 0} 2 x^{3})$

Step 7

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

$\frac{1}{h} (2 {(lim_{x \to 0} x + h)}^{3} - 2 lim_{x \to 0} x^{3})$

Step 8

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

$\frac{1}{h} (2 {(lim_{x \to 0} x + h)}^{3} - 2 {(lim_{x \to 0} x)}^{3})$

Step 9

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

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Step 9.1

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

$\frac{1}{h} (2 {(0 + h)}^{3} - 2 {(lim_{x \to 0} x)}^{3})$

Step 9.2

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

$\frac{1}{h} (2 {(0 + h)}^{3} - 2 \cdot 0^{3})$

Step 10

Simplify the answer.

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Step 10.1

Add $0$ and $h$ .

$\frac{1}{h} (2 h^{3} - 2 \cdot 0^{3})$

Step 10.2

Simplify each term.

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Step 10.2.1

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

$\frac{1}{h} (2 h^{3} - 2 \cdot 0)$

Step 10.2.2

Multiply $- 2$ by $0$ .

$\frac{1}{h} (2 h^{3} + 0)$

Step 10.3

Add $2 h^{3}$ and $0$ .

$\frac{1}{h} (2 h^{3})$

Step 10.4

Rewrite using the commutative property of multiplication.

$2 \frac{1}{h} h^{3}$

Step 10.5

Combine $2$ and $\frac{1}{h}$ .

$\frac{2}{h} h^{3}$

Step 10.6

Cancel the common factor of $h$ .

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Step 10.6.1

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

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

Step 10.6.2

Cancel the common factor.

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

Step 10.6.3

Rewrite the expression.

$2 h^{2}$