Calculus Examples

$lim_{x \to 9} \frac{x - 9}{x^{2} - 18 x + 81}$

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.

$\frac{lim_{x \to 9} x - 9}{lim_{x \to 9} x^{2} - 18 x + 81}$

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 $9$ .

$\frac{lim_{x \to 9} x - lim_{x \to 9} 9}{lim_{x \to 9} x^{2} - 18 x + 81}$

Step 1.1.2.1.2

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

$\frac{lim_{x \to 9} x - 1 \cdot 9}{lim_{x \to 9} x^{2} - 18 x + 81}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{9 - 1 \cdot 9}{lim_{x \to 9} x^{2} - 18 x + 81}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $9$ .

$\frac{9 - 9}{lim_{x \to 9} x^{2} - 18 x + 81}$

Step 1.1.2.3.2

Subtract $9$ from $9$ .

$\frac{0}{lim_{x \to 9} x^{2} - 18 x + 81}$

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 $9$ .

$\frac{0}{lim_{x \to 9} x^{2} - lim_{x \to 9} 18 x + lim_{x \to 9} 81}$

Step 1.1.3.2

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

$\frac{0}{{(lim_{x \to 9} x)}^{2} - lim_{x \to 9} 18 x + lim_{x \to 9} 81}$

Step 1.1.3.3

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{{(lim_{x \to 9} x)}^{2} - 18 lim_{x \to 9} x + lim_{x \to 9} 81}$

Step 1.1.3.4

Evaluate the limit of $81$ which is constant as $x$ approaches $9$ .

$\frac{0}{{(lim_{x \to 9} x)}^{2} - 18 lim_{x \to 9} x + 81}$

Step 1.1.3.5

Evaluate the limits by plugging in $9$ for all occurrences of $x$ .

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Step 1.1.3.5.1

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{0}{9^{2} - 18 lim_{x \to 9} x + 81}$

Step 1.1.3.5.2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\frac{0}{9^{2} - 18 \cdot 9 + 81}$

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 $9$ to the power of $2$ .

$\frac{0}{81 - 18 \cdot 9 + 81}$

Step 1.1.3.6.1.2

Multiply $- 18$ by $9$ .

$\frac{0}{81 - 162 + 81}$

Step 1.1.3.6.2

Subtract $162$ from $81$ .

$\frac{0}{- 81 + 81}$

Step 1.1.3.6.3

Add $- 81$ and $81$ .

$\frac{0}{0}$

Step 1.1.3.6.4

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.

$lim_{x \to 9} \frac{x - 9}{x^{2} - 18 x + 81} = lim_{x \to 9} \frac{\frac{d}{d x} [x - 9]}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 9} \frac{\frac{d}{d x} [x - 9]}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3.2

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

$lim_{x \to 9} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3.3

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

$lim_{x \to 9} \frac{1 + \frac{d}{d x} [- 9]}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3.4

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

$lim_{x \to 9} \frac{1 + 0}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to 9} \frac{1}{\frac{d}{d x} [x^{2} - 18 x + 81]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{2} - 18 x + 81$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]$ .

$lim_{x \to 9} \frac{1}{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]}$

Step 1.3.7

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

$lim_{x \to 9} \frac{1}{2 x + \frac{d}{d x} [- 18 x] + \frac{d}{d x} [81]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [- 18 x]$ .

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Step 1.3.8.1

Since $- 18$ is constant with respect to $x$ , the derivative of $- 18 x$ with respect to $x$ is $- 18 \frac{d}{d x} [x]$ .

$lim_{x \to 9} \frac{1}{2 x - 18 \frac{d}{d x} [x] + \frac{d}{d x} [81]}$

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 $n = 1$ .

$lim_{x \to 9} \frac{1}{2 x - 18 \cdot 1 + \frac{d}{d x} [81]}$

Step 1.3.8.3

Multiply $- 18$ by $1$ .

$lim_{x \to 9} \frac{1}{2 x - 18 + \frac{d}{d x} [81]}$

Step 1.3.9

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$lim_{x \to 9} \frac{1}{2 x - 18 + 0}$

Step 1.3.10

Add $2 x - 18$ and $0$ .

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

Step 2

Since the function approaches $- \infty$ from the left and $\infty$ from the right, the limit does not exist.

$Does not exist$