Calculus Examples

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

Step 1

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

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

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Factor $3$ out of $3 {(0 + h)}^{2} - 3 \cdot 0^{2}$ .

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Step 9.1.1.1

Factor $3$ out of $3 {(0 + h)}^{2}$ .

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

Step 9.1.1.2

Factor $3$ out of $- 3 \cdot 0^{2}$ .

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

Step 9.1.1.3

Factor $3$ out of $3 ({(0 + h)}^{2}) + 3 (- 0^{2})$ .

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

Step 9.1.2

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

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

Step 9.1.3

Simplify.

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Step 9.1.3.1

Add $0 + h$ and $0$ .

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

Step 9.1.3.2

Add $0$ and $h$ .

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

Step 9.1.3.3

Add $0 + h$ and $0$ .

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

Step 9.1.3.4

Add $0$ and $h$ .

$\frac{3 (h \cdot h)}{h}$

$\frac{3 h \cdot h}{h}$

Step 9.1.4

Combine exponents.

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Step 9.1.4.1

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

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

Step 9.1.4.2

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

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

Step 9.1.4.3

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

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

Step 9.1.4.4

Add $1$ and $1$ .

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

Step 9.2

Cancel the common factor of $h^{2}$ and $h$ .

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Step 9.2.1

Factor $h$ out of $3 h^{2}$ .

$\frac{h (3 h)}{h}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

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

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

Step 9.2.2.2

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

$\frac{h (3 h)}{h \cdot 1}$

Step 9.2.2.3

Cancel the common factor.

$\frac{h (3 h)}{h \cdot 1}$

Step 9.2.2.4

Rewrite the expression.

$\frac{3 h}{1}$

Step 9.2.2.5

Divide $3 h$ by $1$ .

$3 h$