Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

$\frac{9 - {(lim_{x \to - 3} x)}^{2}}{lim_{x \to - 3} x^{3} + 3 x^{2}}$

Step 1.1.2.2

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

$\frac{9 - {(- 3)}^{2}}{lim_{x \to - 3} x^{3} + 3 x^{2}}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

$\frac{9 - 1 \cdot 9}{lim_{x \to - 3} x^{3} + 3 x^{2}}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $9$ .

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

Step 1.1.2.3.2

Subtract $9$ from $9$ .

$\frac{0}{lim_{x \to - 3} x^{3} + 3 x^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to - 3} x^{3} + lim_{x \to - 3} 3 x^{2}}$

Step 1.1.3.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to - 3} x)}^{3} + lim_{x \to - 3} 3 x^{2}}$

Step 1.1.3.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to - 3} x)}^{3} + 3 lim_{x \to - 3} x^{2}}$

Step 1.1.3.4

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

$\frac{0}{{(lim_{x \to - 3} x)}^{3} + 3 {(lim_{x \to - 3} x)}^{2}}$

Step 1.1.3.5

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

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Step 1.1.3.5.1

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

$\frac{0}{{(- 3)}^{3} + 3 {(lim_{x \to - 3} x)}^{2}}$

Step 1.1.3.5.2

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

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

Step 1.1.3.6

Simplify the answer.

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Step 1.1.3.6.1

Simplify each term.

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Step 1.1.3.6.1.1

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

$\frac{0}{- 27 + 3 {(- 3)}^{2}}$

Step 1.1.3.6.1.2

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

$\frac{0}{- 27 + 3 \cdot 9}$

Step 1.1.3.6.1.3

Multiply $3$ by $9$ .

$\frac{0}{- 27 + 27}$

Step 1.1.3.6.2

Add $- 27$ and $27$ .

$\frac{0}{0}$

Step 1.1.3.6.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.7

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 1.3.4.2

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

$lim_{x \to - 3} \frac{0 - (2 x)}{\frac{d}{d x} [x^{3} + 3 x^{2}]}$

Step 1.3.4.3

Multiply $2$ by $- 1$ .

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

Step 1.3.5

Subtract $2 x$ from $0$ .

$lim_{x \to - 3} \frac{- 2 x}{\frac{d}{d x} [x^{3} + 3 x^{2}]}$

Step 1.3.6

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

$lim_{x \to - 3} \frac{- 2 x}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}]}$

Step 1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to - 3} \frac{- 2 x}{3 x^{2} + \frac{d}{d x} [3 x^{2}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.3.8.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to - 3} \frac{- 2 x}{3 x^{2} + 3 \frac{d}{d x} [x^{2}]}$

Step 1.3.8.2

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

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

Step 1.3.8.3

Multiply $2$ by $3$ .

$lim_{x \to - 3} \frac{- 2 x}{3 x^{2} + 6 x}$

Step 2

Evaluate the limit.

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Step 2.1

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

$- 2 lim_{x \to - 3} \frac{x}{3 x^{2} + 6 x}$

Step 2.2

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

$- 2 \frac{lim_{x \to - 3} x}{lim_{x \to - 3} 3 x^{2} + 6 x}$

Step 2.3

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

$- 2 \frac{lim_{x \to - 3} x}{lim_{x \to - 3} 3 x^{2} + lim_{x \to - 3} 6 x}$

Step 2.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$- 2 \frac{lim_{x \to - 3} x}{3 lim_{x \to - 3} x^{2} + lim_{x \to - 3} 6 x}$

Step 2.5

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

$- 2 \frac{lim_{x \to - 3} x}{3 {(lim_{x \to - 3} x)}^{2} + lim_{x \to - 3} 6 x}$

Step 2.6

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$- 2 \frac{lim_{x \to - 3} x}{3 {(lim_{x \to - 3} x)}^{2} + 6 lim_{x \to - 3} x}$

Step 3

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

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Step 3.1

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

$- 2 \frac{- 3}{3 {(lim_{x \to - 3} x)}^{2} + 6 lim_{x \to - 3} x}$

Step 3.2

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

$- 2 \frac{- 3}{3 {(- 3)}^{2} + 6 lim_{x \to - 3} x}$

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $- 3$ and $3 {(- 3)}^{2} + 6 \cdot - 3$ .

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Step 4.1.1

Factor $- 3$ out of $- 3$ .

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

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Factor $- 3$ out of $6 \cdot - 3$ .

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

Step 4.1.2.3

Factor $- 3$ out of $- 3 (- {(- 3)}^{2}) - 3 (- 2 \cdot - 3)$ .

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

Step 4.1.2.4

Cancel the common factor.

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

Step 4.1.2.5

Rewrite the expression.

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

$- 2 \frac{1}{- 1 \cdot 9 - 2 \cdot - 3}$

Step 4.2.2

Multiply $- 1$ by $9$ .

$- 2 \frac{1}{- 9 - 2 \cdot - 3}$

Step 4.2.3

Multiply $- 2$ by $- 3$ .

$- 2 \frac{1}{- 9 + 6}$

Step 4.2.4

Add $- 9$ and $6$ .

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

Step 4.3

Move the negative in front of the fraction.

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

Step 4.4

Multiply $- 2 (- \frac{1}{3})$ .

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Step 4.4.1

Multiply $- 1$ by $- 2$ .

$2 (\frac{1}{3})$

Step 4.4.2

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$