Calculus Examples

$lim_{x \to 4} \frac{3 (y^{2} - 1)}{6 y^{2} {(y - 1)}^{3}}$

Step 1

Cancel the common factors.

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Step 1.1

Factor $3$ out of $6 y^{2} {(y - 1)}^{3}$ .

$lim_{x \to 4} \frac{3 (y^{2} - 1)}{3 (2 y^{2} {(y - 1)}^{3})}$

Step 1.2

Cancel the common factor.

$lim_{x \to 4} \frac{3 (y^{2} - 1)}{3 (2 y^{2} {(y - 1)}^{3})}$

Step 1.3

Rewrite the expression.

$lim_{x \to 4} \frac{y^{2} - 1}{2 y^{2} {(y - 1)}^{3}}$

Step 2

Evaluate the limit of $\frac{y^{2} - 1}{2 y^{2} {(y - 1)}^{3}}$ which is constant as $x$ approaches $4$ .

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Cancel the common factor of $y - 1$ and ${(y - 1)}^{3}$ .

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Step 3.2.1

Factor $y - 1$ out of $(y + 1) (y - 1)$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $y - 1$ out of $2 y^{2} {(y - 1)}^{3}$ .

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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