Precalculus Examples

$\frac{lim_{x \to 1} x^{4} - 1}{x^{6} - 1}$

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 $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{4} - lim_{x \to 1} 1}{x^{6} - 1}$

Step 1.2

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

$\frac{{(lim_{x \to 1} x)}^{4} - lim_{x \to 1} 1}{x^{6} - 1}$

Step 1.3

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{4} - 1 \cdot 1}{x^{6} - 1}$

Step 2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{4} - 1 \cdot 1}{x^{6} - 1}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

One to any power is one.

$\frac{1 - 1 \cdot 1}{x^{6} - 1}$

Step 3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{x^{6} - 1}$

Step 3.1.3

Subtract $1$ from $1$ .

$\frac{0}{x^{6} - 1}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$\frac{0}{{(x^{2})}^{3} - 1}$

Step 3.2.2

Rewrite $1$ as $1^{3}$ .

$\frac{0}{{(x^{2})}^{3} - 1^{3}}$

Step 3.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x^{2}$ and $b = 1$ .

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

Step 3.2.4

Simplify.

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Step 3.2.4.1

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

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

Step 3.2.4.2

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 $b = 1$ .

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

Step 3.2.4.3

Multiply $x^{2}$ by $1$ .

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

Step 3.2.5

Simplify each term.

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Step 3.2.5.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.2.5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.5.1.2

Multiply $2$ by $2$ .

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

Step 3.2.5.2

One to any power is one.

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

Step 3.3

Divide $0$ by $(x + 1) (x - 1) (x^{4} + x^{2} + 1)$ .

$0$