Calculus Examples

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

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.

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

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 $- 3$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

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 $2$ .

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $9$ .

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

Step 1.1.2.3.2

Subtract $9$ from $9$ .

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

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 $- 3$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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

Step 1.1.3.6

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

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Step 1.1.3.6.1

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

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

Step 1.1.3.6.2

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

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

Step 1.1.3.7

Simplify the answer.

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Step 1.1.3.7.1

Simplify each term.

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Step 1.1.3.7.1.1

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

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

Step 1.1.3.7.1.2

Multiply $2$ by $9$ .

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

Step 1.1.3.7.1.3

Multiply $7$ by $- 3$ .

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

Step 1.1.3.7.2

Subtract $21$ from $18$ .

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

Step 1.1.3.7.3

Add $- 3$ and $3$ .

$\frac{0}{0}$

Step 1.1.3.7.4

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

Undefined

$\frac{0}{0}$

Step 1.1.3.8

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $2 x$ and $0$ .

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

Step 1.3.6

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

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

Step 1.3.7

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

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Step 1.3.7.1

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

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

Step 1.3.7.2

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

$lim_{x \to - 3} \frac{2 x}{2 (2 x) + \frac{d}{d x} [7 x] + \frac{d}{d x} [3]}$

Step 1.3.7.3

Multiply $2$ by $2$ .

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

Step 1.3.8

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

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Step 1.3.8.1

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

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

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 $n = 1$ .

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

Step 1.3.8.3

Multiply $7$ by $1$ .

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

Step 1.3.9

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

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

Step 1.3.10

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

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

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 $x$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

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 $x$ .

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $4$ by $- 3$ .

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

Step 4.1.2

Add $- 12$ and $7$ .

$2 (\frac{- 3}{- 5})$

Step 4.2

Dividing two negative values results in a positive value.

$2 (\frac{3}{5})$

Step 4.3

Multiply $2 (\frac{3}{5})$ .

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Step 4.3.1

Combine $2$ and $\frac{3}{5}$ .

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

Step 4.3.2

Multiply $2$ by $3$ .

$\frac{6}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{6}{5}$

Decimal Form:

$1.2$