Calculus Examples

$lim_{x \to 3} \frac{7 (y^{2} - 1)}{4 y^{2} {(y - 1)}^{3}}$

Step 1

Evaluate the limit of $\frac{7 (y^{2} - 1)}{4 y^{2} {(y - 1)}^{3}}$ which is constant as $x$ approaches $3$ .

$\frac{7 (y^{2} - 1)}{4 y^{2} {(y - 1)}^{3}}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\frac{7 (y^{2} - 1^{2})}{4 y^{2} {(y - 1)}^{3}}$

Step 2.1.2

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 $b = 1$ .

$\frac{7 (y + 1) (y - 1)}{4 y^{2} {(y - 1)}^{3}}$

Step 2.2

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

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Step 2.2.1

Factor $y - 1$ out of $7 (y + 1) (y - 1)$ .

$\frac{(y - 1) (7 (y + 1))}{4 y^{2} {(y - 1)}^{3}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $y - 1$ out of $4 y^{2} {(y - 1)}^{3}$ .

$\frac{(y - 1) (7 (y + 1))}{(y - 1) (4 y^{2} {(y - 1)}^{2})}$

Step 2.2.2.2

Cancel the common factor.

$\frac{(y - 1) (7 (y + 1))}{(y - 1) (4 y^{2} {(y - 1)}^{2})}$

Step 2.2.2.3

Rewrite the expression.

$\frac{7 (y + 1)}{4 y^{2} {(y - 1)}^{2}}$