Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

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

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

Step 1.1.2.6

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

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Step 1.1.2.6.1

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

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

Step 1.1.2.6.2

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

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

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

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

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

Step 1.1.2.7.1.2

Multiply $2$ by $9$ .

$\frac{18 - 3 \cdot 3 - 1 \cdot 9}{lim_{x \to 3} x^{2} - 9}$

Step 1.1.2.7.1.3

Multiply $- 3$ by $3$ .

$\frac{18 - 9 - 1 \cdot 9}{lim_{x \to 3} x^{2} - 9}$

Step 1.1.2.7.1.4

Multiply $- 1$ by $9$ .

$\frac{18 - 9 - 9}{lim_{x \to 3} x^{2} - 9}$

Step 1.1.2.7.2

Subtract $9$ from $18$ .

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

Step 1.1.2.7.3

Subtract $9$ from $9$ .

$\frac{0}{lim_{x \to 3} x^{2} - 9}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to 3} x^{2} - lim_{x \to 3} 9}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to 3} x)}^{2} - lim_{x \to 3} 9}$

Step 1.1.3.1.3

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

$\frac{0}{{(lim_{x \to 3} x)}^{2} - 1 \cdot 9}$

Step 1.1.3.2

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

$\frac{0}{3^{2} - 1 \cdot 9}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

$\frac{0}{9 - 1 \cdot 9}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $9$ .

$\frac{0}{9 - 9}$

Step 1.1.3.3.2

Subtract $9$ from $9$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $2 x^{2} - 3 x - 9$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 9]$ .

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

Step 1.3.3.2

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

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

Step 1.3.3.3

Multiply $2$ by $2$ .

$lim_{x \to 3} \frac{4 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 9]}$

Step 1.3.4

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

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Step 1.3.4.1

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

$lim_{x \to 3} \frac{4 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 9]}$

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 $n = 1$ .

$lim_{x \to 3} \frac{4 x - 3 \cdot 1 + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 9]}$

Step 1.3.4.3

Multiply $- 3$ by $1$ .

$lim_{x \to 3} \frac{4 x - 3 + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 9]}$

Step 1.3.5

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

$lim_{x \to 3} \frac{4 x - 3 + 0}{\frac{d}{d x} [x^{2} - 9]}$

Step 1.3.6

Add $4 x - 3$ and $0$ .

$lim_{x \to 3} \frac{4 x - 3}{\frac{d}{d x} [x^{2} - 9]}$

Step 1.3.7

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

$lim_{x \to 3} \frac{4 x - 3}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]}$

Step 1.3.8

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

$lim_{x \to 3} \frac{4 x - 3}{2 x + \frac{d}{d x} [- 9]}$

Step 1.3.9

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

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

Step 1.3.10

Add $2 x$ and $0$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 3} \frac{4 x - 3}{x}$

Step 2.2

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

$\frac{1}{2} \cdot \frac{lim_{x \to 3} 4 x - 3}{lim_{x \to 3} x}$

Step 2.3

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

$\frac{1}{2} \cdot \frac{lim_{x \to 3} 4 x - lim_{x \to 3} 3}{lim_{x \to 3} x}$

Step 2.4

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

$\frac{1}{2} \cdot \frac{4 lim_{x \to 3} x - lim_{x \to 3} 3}{lim_{x \to 3} x}$

Step 2.5

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

$\frac{1}{2} \cdot \frac{4 lim_{x \to 3} x - 1 \cdot 3}{lim_{x \to 3} x}$

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 $x$ .

$\frac{1}{2} \cdot \frac{4 \cdot 3 - 1 \cdot 3}{lim_{x \to 3} x}$

Step 3.2

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

$\frac{1}{2} \cdot \frac{4 \cdot 3 - 1 \cdot 3}{3}$

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $4 \cdot 3 - 1 \cdot 3$ and $3$ .

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Step 4.1.1

Factor $- 1$ out of $4 \cdot 3$ .

$\frac{1}{2} \cdot \frac{- (- 4 \cdot 3) - 1 \cdot 3}{3}$

Step 4.1.2

Factor $- 1$ out of $- 1 \cdot 3$ .

$\frac{1}{2} \cdot \frac{- (- 4 \cdot 3) - 1 (3)}{3}$

Step 4.1.3

Factor $- 1$ out of $- (- 4 \cdot 3) - 1 (3)$ .

$\frac{1}{2} \cdot \frac{- (- 4 \cdot 3 + 3)}{3}$

Step 4.1.4

Rewrite $- (- 4 \cdot 3 + 3)$ as $- 1 (- 4 \cdot 3 + 3)$ .

$\frac{1}{2} \cdot \frac{- 1 (- 4 \cdot 3 + 3)}{3}$

Step 4.1.5

Factor $3$ out of $- 1 (- 4 \cdot 3 + 3)$ .

$\frac{1}{2} \cdot \frac{3 (- 1 (- 4 + 1))}{3}$

Step 4.1.6

Cancel the common factors.

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Step 4.1.6.1

Factor $3$ out of $3$ .

$\frac{1}{2} \cdot \frac{3 (- 1 (- 4 + 1))}{3 (1)}$

Step 4.1.6.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{3 (- 1 (- 4 + 1))}{3 \cdot 1}$

Step 4.1.6.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{- 1 (- 4 + 1)}{1}$

Step 4.1.6.4

Divide $- 1 (- 4 + 1)$ by $1$ .

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

Step 4.2

Rewrite $- 1 (- 4 + 1)$ as $- (- 4 + 1)$ .

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

Step 4.3

Add $- 4$ and $1$ .

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

Step 4.4

Multiply $- 1$ by $- 3$ .

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

Step 4.5

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

$\frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$