Calculus Examples

$lim_{x \to 0} \frac{1 + x^{6} - 1}{1 + x^{2} - 1}$

Step 1

Evaluate the limit.

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Step 1.1

Simplify the limit argument.

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Step 1.1.1

Combine terms.

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Step 1.1.1.1

Subtract $1$ from $1$ .

$lim_{x \to 0} \frac{x^{6} + 0}{1 + x^{2} - 1}$

Step 1.1.1.2

Add $x^{6}$ and $0$ .

$lim_{x \to 0} \frac{x^{6}}{1 + x^{2} - 1}$

Step 1.1.1.3

Subtract $1$ from $1$ .

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

Step 1.1.1.4

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

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

Step 1.1.2

Cancel the common factor of $x^{6}$ and $x^{2}$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

Cancel the common factors.

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Step 1.1.2.2.1

Multiply by $1$ .

$lim_{x \to 0} \frac{x^{2} x^{4}}{x^{2} \cdot 1}$

Step 1.1.2.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{x^{2} x^{4}}{x^{2} \cdot 1}$

Step 1.1.2.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{x^{4}}{1}$

Step 1.1.2.2.4

Divide $x^{4}$ by $1$ .

$lim_{x \to 0} x^{4}$

Step 1.2

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

${(lim_{x \to 0} x)}^{4}$

Step 2

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

$0^{4}$

Step 3

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

$0$