Calculus Examples

$lim_{x \to - 3} \sqrt{\frac{y^{2} - 9}{2 y^{2} + 7 y + 3}}$

Step 1

Evaluate the limit of $\sqrt{\frac{y^{2} - 9}{2 y^{2} + 7 y + 3}}$ which is constant as $x$ approaches $- 3$ .

$\sqrt{\frac{y^{2} - 9}{2 y^{2} + 7 y + 3}}$

Step 2

Simplify the answer.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{y^{2} - 3^{2}}{2 y^{2} + 7 y + 3}}$

Step 2.2

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 $b = 3$ .

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

Step 2.3

Factor by grouping.

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Step 2.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = 7$ .

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Step 2.3.1.1

Factor $7$ out of $7 y$ .

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

Step 2.3.1.2

Rewrite $7$ as $1$ plus $6$

$\sqrt{\frac{(y + 3) (y - 3)}{2 y^{2} + (1 + 6) y + 3}}$

Step 2.3.1.3

Apply the distributive property.

$\sqrt{\frac{(y + 3) (y - 3)}{2 y^{2} + 1 y + 6 y + 3}}$

Step 2.3.1.4

Multiply $y$ by $1$ .

$\sqrt{\frac{(y + 3) (y - 3)}{2 y^{2} + y + 6 y + 3}}$

Step 2.3.2

Factor out the greatest common factor from each group.

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Step 2.3.2.1

Group the first two terms and the last two terms.

$\sqrt{\frac{(y + 3) (y - 3)}{(2 y^{2} + y) + 6 y + 3}}$

Step 2.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.3.3

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

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

Step 2.4

Reduce the expression $\frac{(y + 3) (y - 3)}{(2 y + 1) (y + 3)}$ by cancelling the common factors.

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$\sqrt{\frac{y - 3}{2 y + 1}}$

Step 2.5

Rewrite $\sqrt{\frac{y - 3}{2 y + 1}}$ as $\frac{\sqrt{y - 3}}{\sqrt{2 y + 1}}$ .

$\frac{\sqrt{y - 3}}{\sqrt{2 y + 1}}$

Step 2.6

Multiply $\frac{\sqrt{y - 3}}{\sqrt{2 y + 1}}$ by $\frac{\sqrt{2 y + 1}}{\sqrt{2 y + 1}}$ .

$\frac{\sqrt{y - 3}}{\sqrt{2 y + 1}} \cdot \frac{\sqrt{2 y + 1}}{\sqrt{2 y + 1}}$

Step 2.7

Combine and simplify the denominator.

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Step 2.7.1

Multiply $\frac{\sqrt{y - 3}}{\sqrt{2 y + 1}}$ by $\frac{\sqrt{2 y + 1}}{\sqrt{2 y + 1}}$ .

$\frac{\sqrt{y - 3} \sqrt{2 y + 1}}{\sqrt{2 y + 1} \sqrt{2 y + 1}}$

Step 2.7.2

Raise $\sqrt{2 y + 1}$ to the power of $1$ .

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

Step 2.7.3

Raise $\sqrt{2 y + 1}$ to the power of $1$ .

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

Step 2.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.7.5

Add $1$ and $1$ .

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

Step 2.7.6

Rewrite ${\sqrt{2 y + 1}}^{2}$ as $2 y + 1$ .

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Step 2.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 y + 1}$ as ${(2 y + 1)}^{\frac{1}{2}}$ .

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

Step 2.7.6.2

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

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

Step 2.7.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.7.6.4

Cancel the common factor of $2$ .

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Step 2.7.6.4.1

Cancel the common factor.

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

Step 2.7.6.4.2

Rewrite the expression.

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

Step 2.7.6.5

Simplify.

$\frac{\sqrt{y - 3} \sqrt{2 y + 1}}{2 y + 1}$

Step 2.8

Combine using the product rule for radicals.

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