Calculus Examples

$lim_{x \to - 2} \frac{w^{3} - w}{2 w^{2} + 3 w}$

Step 1

Evaluate the limit of $\frac{w^{3} - w}{2 w^{2} + 3 w}$ which is constant as $x$ approaches $- 2$ .

$\frac{w^{3} - w}{2 w^{2} + 3 w}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Factor $w$ out of $w^{3} - w$ .

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Step 2.1.1.1

Factor $w$ out of $w^{3}$ .

$\frac{w \cdot w^{2} - w}{2 w^{2} + 3 w}$

Step 2.1.1.2

Factor $w$ out of $- w$ .

$\frac{w \cdot w^{2} + w \cdot - 1}{2 w^{2} + 3 w}$

Step 2.1.1.3

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

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

Step 2.1.2

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

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

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 $a = w$ and $b = 1$ .

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

Step 2.2

Factor $w$ out of $2 w^{2} + 3 w$ .

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Step 2.2.1

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

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

Step 2.2.2

Factor $w$ out of $3 w$ .

$\frac{w (w + 1) (w - 1)}{w (2 w) + w \cdot 3}$

Step 2.2.3

Factor $w$ out of $w (2 w) + w \cdot 3$ .

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

Step 2.3

Cancel the common factor of $w$ .

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Step 2.3.1

Cancel the common factor.

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

Step 2.3.2

Rewrite the expression.

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