Calculus Examples

$\frac{lim_{x \to 1} u^{4} - 1}{u^{3} - 1}$

Step 1

Evaluate the limit of $u^{4} - 1$ which is constant as $x$ approaches $1$ .

$\frac{u^{4} - 1}{u^{3} - 1}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

$\frac{(u^{2} + 1) (u^{2} - 1)}{u^{3} - 1}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

$\frac{(u^{2} + 1) (u^{2} - 1^{2})}{u^{3} - 1}$

Step 2.1.4.2

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

$\frac{(u^{2} + 1) (u + 1) (u - 1)}{u^{3} - 1}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

$\frac{(u^{2} + 1) (u + 1) (u - 1)}{u^{3} - 1^{3}}$

Step 2.2.2

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 $a = u$ and $b = 1$ .

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $u$ by $1$ .

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

Step 2.2.3.2

One to any power is one.

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

Step 2.3

Cancel the common factor of $u - 1$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

$\frac{(u^{2} + 1) (u + 1)}{u^{2} + u + 1}$