Calculus Examples

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

Step 1

Combine fractions using a common denominator.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

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

Step 1.2.2

Simplify.

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Step 1.2.2.1

Add $x$ and $x$ .

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

Step 1.2.2.2

Subtract $x$ from $x$ .

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

Step 1.2.2.3

Add $h$ and $0$ .

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

Step 1.3

Cancel the common factor of $h$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Divide $h + 2 x$ by $1$ .

$lim_{h \to 0} h + 2 x$

Step 2

Evaluate the limit.

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Step 2.1

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

$lim_{h \to 0} h + lim_{h \to 0} 2 x$

Step 2.2

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

$lim_{h \to 0} h + 2 x$

Step 3

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

$0 + 2 x$

Step 4

Add $0$ and $2 x$ .

$2 x$