Calculus Examples

$lim_{x \to 12} \frac{x - 12}{x^{2} - 144}$

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.

$\frac{lim_{x \to 12} x - 12}{lim_{x \to 12} x^{2} - 144}$

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 $12$ .

$\frac{lim_{x \to 12} x - lim_{x \to 12} 12}{lim_{x \to 12} x^{2} - 144}$

Step 1.1.2.1.2

Evaluate the limit of $12$ which is constant as $x$ approaches $12$ .

$\frac{lim_{x \to 12} x - 1 \cdot 12}{lim_{x \to 12} x^{2} - 144}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $12$ for $x$ .

$\frac{12 - 1 \cdot 12}{lim_{x \to 12} x^{2} - 144}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $12$ .

$\frac{12 - 12}{lim_{x \to 12} x^{2} - 144}$

Step 1.1.2.3.2

Subtract $12$ from $12$ .

$\frac{0}{lim_{x \to 12} x^{2} - 144}$

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 $12$ .

$\frac{0}{lim_{x \to 12} x^{2} - lim_{x \to 12} 144}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to 12} x)}^{2} - lim_{x \to 12} 144}$

Step 1.1.3.1.3

Evaluate the limit of $144$ which is constant as $x$ approaches $12$ .

$\frac{0}{{(lim_{x \to 12} x)}^{2} - 1 \cdot 144}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $12$ for $x$ .

$\frac{0}{12^{2} - 1 \cdot 144}$

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 $12$ to the power of $2$ .

$\frac{0}{144 - 1 \cdot 144}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $144$ .

$\frac{0}{144 - 144}$

Step 1.1.3.3.2

Subtract $144$ from $144$ .

$\frac{0}{0}$

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.

$lim_{x \to 12} \frac{x - 12}{x^{2} - 144} = lim_{x \to 12} \frac{\frac{d}{d x} [x - 12]}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 12} \frac{\frac{d}{d x} [x - 12]}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3.2

By the Sum Rule, the derivative of $x - 12$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 12]$ .

$lim_{x \to 12} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 12]}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3.3

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

$lim_{x \to 12} \frac{1 + \frac{d}{d x} [- 12]}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3.4

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$lim_{x \to 12} \frac{1 + 0}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to 12} \frac{1}{\frac{d}{d x} [x^{2} - 144]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} - 144$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 144]$ .

$lim_{x \to 12} \frac{1}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 144]}$

Step 1.3.7

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

$lim_{x \to 12} \frac{1}{2 x + \frac{d}{d x} [- 144]}$

Step 1.3.8

Since $- 144$ is constant with respect to $x$ , the derivative of $- 144$ with respect to $x$ is $0$ .

$lim_{x \to 12} \frac{1}{2 x + 0}$

Step 1.3.9

Add $2 x$ and $0$ .

$lim_{x \to 12} \frac{1}{2 x}$

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 $x$ .

$\frac{1}{2} lim_{x \to 12} \frac{1}{x}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $12$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 12} 1}{lim_{x \to 12} x}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $12$ .

$\frac{1}{2} \cdot \frac{1}{lim_{x \to 12} x}$

Step 3

Evaluate the limit of $x$ by plugging in $12$ for $x$ .

$\frac{1}{2} \cdot \frac{1}{12}$

Step 4

Multiply $\frac{1}{2} \cdot \frac{1}{12}$ .

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Step 4.1

Multiply $\frac{1}{2}$ by $\frac{1}{12}$ .

$\frac{1}{2 \cdot 12}$

Step 4.2

Multiply $2$ by $12$ .

$\frac{1}{24}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{24}$

Decimal Form:

$0.041\overline{6}$