Calculus Examples

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

Step 1

Move the limit under the radical sign.

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 2.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- 3$ .

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

Step 2.1.2.1.2

Move the exponent $2$ from $y^{2}$ outside the limit using the Limits Power Rule.

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

Step 2.1.2.1.3

Evaluate the limit of $9$ which is constant as $y$ approaches $- 3$ .

$\sqrt{\frac{{(lim_{y \to - 3} y)}^{2} - 1 \cdot 9}{lim_{y \to - 3} 2 y^{2} + 7 y + 3}}$

Step 2.1.2.2

Evaluate the limit of $y$ by plugging in $- 3$ for $y$ .

$\sqrt{\frac{{(- 3)}^{2} - 1 \cdot 9}{lim_{y \to - 3} 2 y^{2} + 7 y + 3}}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Raise $- 3$ to the power of $2$ .

$\sqrt{\frac{9 - 1 \cdot 9}{lim_{y \to - 3} 2 y^{2} + 7 y + 3}}$

Step 2.1.2.3.1.2

Multiply $- 1$ by $9$ .

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

Step 2.1.2.3.2

Subtract $9$ from $9$ .

$\sqrt{\frac{0}{lim_{y \to - 3} 2 y^{2} + 7 y + 3}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- 3$ .

$\sqrt{\frac{0}{lim_{y \to - 3} 2 y^{2} + lim_{y \to - 3} 7 y + lim_{y \to - 3} 3}}$

Step 2.1.3.2

Move the term $2$ outside of the limit because it is constant with respect to $y$ .

$\sqrt{\frac{0}{2 lim_{y \to - 3} y^{2} + lim_{y \to - 3} 7 y + lim_{y \to - 3} 3}}$

Step 2.1.3.3

Move the exponent $2$ from $y^{2}$ outside the limit using the Limits Power Rule.

$\sqrt{\frac{0}{2 {(lim_{y \to - 3} y)}^{2} + lim_{y \to - 3} 7 y + lim_{y \to - 3} 3}}$

Step 2.1.3.4

Move the term $7$ outside of the limit because it is constant with respect to $y$ .

$\sqrt{\frac{0}{2 {(lim_{y \to - 3} y)}^{2} + 7 lim_{y \to - 3} y + lim_{y \to - 3} 3}}$

Step 2.1.3.5

Evaluate the limit of $3$ which is constant as $y$ approaches $- 3$ .

$\sqrt{\frac{0}{2 {(lim_{y \to - 3} y)}^{2} + 7 lim_{y \to - 3} y + 3}}$

Step 2.1.3.6

Evaluate the limits by plugging in $- 3$ for all occurrences of $y$ .

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Step 2.1.3.6.1

Evaluate the limit of $y$ by plugging in $- 3$ for $y$ .

$\sqrt{\frac{0}{2 {(- 3)}^{2} + 7 lim_{y \to - 3} y + 3}}$

Step 2.1.3.6.2

Evaluate the limit of $y$ by plugging in $- 3$ for $y$ .

$\sqrt{\frac{0}{2 {(- 3)}^{2} + 7 \cdot - 3 + 3}}$

Step 2.1.3.7

Simplify the answer.

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Step 2.1.3.7.1

Simplify each term.

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Step 2.1.3.7.1.1

Raise $- 3$ to the power of $2$ .

$\sqrt{\frac{0}{2 \cdot 9 + 7 \cdot - 3 + 3}}$

Step 2.1.3.7.1.2

Multiply $2$ by $9$ .

$\sqrt{\frac{0}{18 + 7 \cdot - 3 + 3}}$

Step 2.1.3.7.1.3

Multiply $7$ by $- 3$ .

$\sqrt{\frac{0}{18 - 21 + 3}}$

Step 2.1.3.7.2

Subtract $21$ from $18$ .

$\sqrt{\frac{0}{- 3 + 3}}$

Step 2.1.3.7.3

Add $- 3$ and $3$ .

$\sqrt{\frac{0}{0}}$

Step 2.1.3.7.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\sqrt{\frac{0}{0}}$

Step 2.1.3.8

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\sqrt{\frac{0}{0}}$

Step 2.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\sqrt{\frac{0}{0}}$

Step 2.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{y \to - 3} \frac{y^{2} - 9}{2 y^{2} + 7 y + 3} = lim_{y \to - 3} \frac{\frac{d}{d y} [y^{2} - 9]}{\frac{d}{d y} [2 y^{2} + 7 y + 3]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

By the Sum Rule, the derivative of $y^{2} - 9$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 9]$ .

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

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

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

Step 2.3.4

Since $- 9$ is constant with respect to $y$ , the derivative of $- 9$ with respect to $y$ is $0$ .

$\sqrt{lim_{y \to - 3} \frac{2 y + 0}{\frac{d}{d y} [2 y^{2} + 7 y + 3]}}$

Step 2.3.5

Add $2 y$ and $0$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{\frac{d}{d y} [2 y^{2} + 7 y + 3]}}$

Step 2.3.6

By the Sum Rule, the derivative of $2 y^{2} + 7 y + 3$ with respect to $y$ is $\frac{d}{d y} [2 y^{2}] + \frac{d}{d y} [7 y] + \frac{d}{d y} [3]$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{\frac{d}{d y} [2 y^{2}] + \frac{d}{d y} [7 y] + \frac{d}{d y} [3]}}$

Step 2.3.7

Evaluate $\frac{d}{d y} [2 y^{2}]$ .

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Step 2.3.7.1

Since $2$ is constant with respect to $y$ , the derivative of $2 y^{2}$ with respect to $y$ is $2 \frac{d}{d y} [y^{2}]$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{2 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [7 y] + \frac{d}{d y} [3]}}$

Step 2.3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{2 \cdot 2 y + \frac{d}{d y} [7 y] + \frac{d}{d y} [3]}}$

Step 2.3.7.3

Multiply $2$ by $2$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + \frac{d}{d y} [7 y] + \frac{d}{d y} [3]}}$

Step 2.3.8

Evaluate $\frac{d}{d y} [7 y]$ .

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Step 2.3.8.1

Since $7$ is constant with respect to $y$ , the derivative of $7 y$ with respect to $y$ is $7 \frac{d}{d y} [y]$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + 7 \frac{d}{d y} [y] + \frac{d}{d y} [3]}}$

Step 2.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + 7 \cdot 1 + \frac{d}{d y} [3]}}$

Step 2.3.8.3

Multiply $7$ by $1$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + 7 + \frac{d}{d y} [3]}}$

Step 2.3.9

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + 7 + 0}}$

Step 2.3.10

Add $4 y + 7$ and $0$ .

$\sqrt{lim_{y \to - 3} \frac{2 y}{4 y + 7}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $2$ outside of the limit because it is constant with respect to $y$ .

$\sqrt{2 lim_{y \to - 3} \frac{y}{4 y + 7}}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $- 3$ .

$\sqrt{2 \frac{lim_{y \to - 3} y}{lim_{y \to - 3} 4 y + 7}}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $y$ approaches $- 3$ .

$\sqrt{2 \frac{lim_{y \to - 3} y}{lim_{y \to - 3} 4 y + lim_{y \to - 3} 7}}$

Step 3.4

Move the term $4$ outside of the limit because it is constant with respect to $y$ .

$\sqrt{2 \frac{lim_{y \to - 3} y}{4 lim_{y \to - 3} y + lim_{y \to - 3} 7}}$

Step 3.5

Evaluate the limit of $7$ which is constant as $y$ approaches $- 3$ .

$\sqrt{2 \frac{lim_{y \to - 3} y}{4 lim_{y \to - 3} y + 7}}$

Step 4

Evaluate the limits by plugging in $- 3$ for all occurrences of $y$ .

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Step 4.1

Evaluate the limit of $y$ by plugging in $- 3$ for $y$ .

$\sqrt{2 \frac{- 3}{4 lim_{y \to - 3} y + 7}}$

Step 4.2

Evaluate the limit of $y$ by plugging in $- 3$ for $y$ .

$\sqrt{2 \frac{- 3}{4 \cdot - 3 + 7}}$

Step 5

Simplify the answer.

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Step 5.1

Combine $2$ and $\frac{- 3}{4 \cdot - 3 + 7}$ .

$\sqrt{\frac{2 \cdot - 3}{4 \cdot - 3 + 7}}$

Step 5.2

Multiply $4$ by $- 3$ .

$\sqrt{\frac{2 \cdot - 3}{- 12 + 7}}$

Step 5.3

Add $- 12$ and $7$ .

$\sqrt{\frac{2 \cdot - 3}{- 5}}$

Step 5.4

Multiply $2$ by $- 3$ .

$\sqrt{\frac{- 6}{- 5}}$

Step 5.5

Dividing two negative values results in a positive value.

$\sqrt{\frac{6}{5}}$

Step 5.6

Rewrite $\sqrt{\frac{6}{5}}$ as $\frac{\sqrt{6}}{\sqrt{5}}$ .

$\frac{\sqrt{6}}{\sqrt{5}}$

Step 5.7

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.8

Combine and simplify the denominator.

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Step 5.8.1

Multiply $\frac{\sqrt{6}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{6} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.8.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 5.8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 5.8.4

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

$\frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.8.5

Add $1$ and $1$ .

$\frac{\sqrt{6} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{6} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.8.6.2

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

$\frac{\sqrt{6} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.8.6.3

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

$\frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.8.6.4

Cancel the common factor of $2$ .

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Step 5.8.6.4.1

Cancel the common factor.

$\frac{\sqrt{6} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.8.6.4.2

Rewrite the expression.

$\frac{\sqrt{6} \sqrt{5}}{5^{1}}$

Step 5.8.6.5

Evaluate the exponent.

$\frac{\sqrt{6} \sqrt{5}}{5}$

Step 5.9

Simplify the numerator.

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Step 5.9.1

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 5}}{5}$

Step 5.9.2

Multiply $6$ by $5$ .

$\frac{\sqrt{30}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{30}}{5}$

Decimal Form:

$1.09544511 \dots$