Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

Add $x$ and $0$ .

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

Step 3.1.2.2

Add $x$ and $x$ .

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

Step 3.1.2.3

Add $x$ and $0$ .

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

Step 3.1.2.4

Subtract $x$ from $x$ .

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

Step 3.1.2.5

Combine exponents.

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Step 3.1.2.5.1

Multiply $0$ by $2$ .

$\frac{0 x}{h}$

Step 3.1.2.5.2

Multiply $0$ by $x$ .

$\frac{0}{h}$

Step 3.2

Divide $0$ by $h$ .

$0$