Calculus Examples

$\frac{lim_{x \to 0} x^{2} + x^{4}}{x^{6} - x^{2}}$

Step 1

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

$\frac{lim_{x \to 0} x^{2} + lim_{x \to 0} x^{4}}{x^{6} - x^{2}}$

Step 2

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

$\frac{{(lim_{x \to 0} x)}^{2} + lim_{x \to 0} x^{4}}{x^{6} - x^{2}}$

Step 3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} x)}^{2} + {(lim_{x \to 0} x)}^{4}}{x^{6} - x^{2}}$

Step 4

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

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Step 4.1

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

$\frac{0^{2} + {(lim_{x \to 0} x)}^{4}}{x^{6} - x^{2}}$

Step 4.2

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

$\frac{0^{2} + 0^{4}}{x^{6} - x^{2}}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$\frac{0 + 0^{4}}{x^{6} - x^{2}}$

Step 5.1.2

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

$\frac{0 + 0}{x^{6} - x^{2}}$

Step 5.1.3

Add $0$ and $0$ .

$\frac{0}{x^{6} - x^{2}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Factor $x^{2}$ out of $x^{6} - x^{2}$ .

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Step 5.2.1.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{0}{x^{2} x^{4} - x^{2}}$

Step 5.2.1.2

Factor $x^{2}$ out of $- x^{2}$ .

$\frac{0}{x^{2} x^{4} + x^{2} \cdot - 1}$

Step 5.2.1.3

Factor $x^{2}$ out of $x^{2} x^{4} + x^{2} \cdot - 1$ .

$\frac{0}{x^{2} (x^{4} - 1)}$

Step 5.2.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 5.2.3

Rewrite $1$ as $1^{2}$ .

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

Step 5.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

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

Step 5.2.5

Simplify.

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Step 5.2.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.2.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

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

Step 5.3

Divide $0$ by $x^{2} (x^{2} + 1) (x + 1) (x - 1)$ .

$0$