Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 8

Simplify the answer.

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Step 8.1

Add $0$ and $h$ .

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

Step 8.2

Simplify each term.

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Step 8.2.1

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

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

Step 8.2.2

Multiply $- 1$ by $0$ .

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

Step 8.3

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

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

Step 8.4

Cancel the common factor of $h$ .

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Step 8.4.1

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

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

$h$