Precalculus Examples

$lim_{x \to 0} \frac{(\frac{1}{6 - x}) - (\frac{1}{6})}{x}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{6 - x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$lim_{x \to 0} \frac{\frac{1}{6 - x} \cdot \frac{6}{6} - (\frac{1}{6})}{x}$

Step 1.2

To write $- (\frac{1}{6})$ as a fraction with a common denominator, multiply by $\frac{6 - x}{6 - x}$ .

$lim_{x \to 0} \frac{\frac{1}{6 - x} \cdot \frac{6}{6} - (\frac{1}{6}) \cdot \frac{6 - x}{6 - x}}{x}$

Step 1.3

Write each expression with a common denominator of $(6 - x) \cdot 6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{6 - x}$ by $\frac{6}{6}$ .

$lim_{x \to 0} \frac{\frac{6}{(6 - x) \cdot 6} - (\frac{1}{6}) \cdot \frac{6 - x}{6 - x}}{x}$

Step 1.3.2

Multiply $\frac{1}{6}$ by $\frac{6 - x}{6 - x}$ .

$lim_{x \to 0} \frac{\frac{6}{(6 - x) \cdot 6} - \frac{6 - x}{6 (6 - x)}}{x}$

Step 1.3.3

Reorder the factors of $(6 - x) \cdot 6$ .

$lim_{x \to 0} \frac{\frac{6}{6 (6 - x)} - \frac{6 - x}{6 (6 - x)}}{x}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{6 - (6 - x)}{6 (6 - x)}}{x}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{6 - (6 - x)}{6 (6 - x)} \cdot \frac{1}{x}$

Step 2.1.2

Multiply $\frac{6 - (6 - x)}{6 (6 - x)}$ by $\frac{1}{x}$ .

$lim_{x \to 0} \frac{6 - (6 - x)}{6 (6 - x) x}$

Step 2.2

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{6} lim_{x \to 0} \frac{6 - (6 - x)}{(6 - x) x}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$\frac{1}{6} \cdot \frac{lim_{x \to 0} 6 - (6 - x)}{lim_{x \to 0} (6 - x) x}$

Step 3.1.2

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.2.1

Evaluate the limit of $6 - (6 - x)$ by plugging in $0$ for $x$ .

$\frac{1}{6} \cdot \frac{6 - (6 + 0)}{lim_{x \to 0} (6 - x) x}$

Step 3.1.2.2

Simplify each term.

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Step 3.1.2.2.1

Add $6$ and $0$ .

$\frac{1}{6} \cdot \frac{6 - 1 \cdot 6}{lim_{x \to 0} (6 - x) x}$

Step 3.1.2.2.2

Multiply $- 1$ by $6$ .

$\frac{1}{6} \cdot \frac{6 - 6}{lim_{x \to 0} (6 - x) x}$

Step 3.1.2.3

Subtract $6$ from $6$ .

$\frac{1}{6} \cdot \frac{0}{lim_{x \to 0} (6 - x) x}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{0}{lim_{x \to 0} 6 - x \cdot lim_{x \to 0} x}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{0}{(lim_{x \to 0} 6 - lim_{x \to 0} x) \cdot lim_{x \to 0} x}$

Step 3.1.3.3

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{0}{(6 - lim_{x \to 0} x) \cdot lim_{x \to 0} x}$

Step 3.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.3.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{6} \cdot \frac{0}{(6 + 0) \cdot lim_{x \to 0} x}$

Step 3.1.3.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{6} \cdot \frac{0}{(6 + 0) \cdot 0}$

Step 3.1.3.5

Simplify the answer.

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Step 3.1.3.5.1

Add $6$ and $0$ .

$\frac{1}{6} \cdot \frac{0}{6 \cdot 0}$

Step 3.1.3.5.2

Multiply $6$ by $0$ .

$\frac{1}{6} \cdot \frac{0}{0}$

Step 3.1.3.5.3

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

Undefined

$\frac{1}{6} \cdot \frac{0}{0}$

Step 3.1.3.6

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

Undefined

$\frac{1}{6} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{6} \cdot \frac{0}{0}$

Step 3.2

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.

$lim_{x \to 0} \frac{6 - (6 - x)}{(6 - x) x} = lim_{x \to 0} \frac{\frac{d}{d x} [6 - (6 - x)]}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{1}{6} lim_{x \to 0} \frac{\frac{d}{d x} [6 - (6 - x)]}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.2

By the Sum Rule, the derivative of $6 - (6 - x)$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- (6 - x)]$ .

$\frac{1}{6} lim_{x \to 0} \frac{\frac{d}{d x} [6] + \frac{d}{d x} [- (6 - x)]}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 + \frac{d}{d x} [- (6 - x)]}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [- (6 - x)]$ .

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Step 3.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (6 - x)$ with respect to $x$ is $- \frac{d}{d x} [6 - x]$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 - \frac{d}{d x} [6 - x]}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.2

By the Sum Rule, the derivative of $6 - x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x]$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 - (\frac{d}{d x} [6] + \frac{d}{d x} [- x])}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 - (0 + \frac{d}{d x} [- x])}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.4

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

$\frac{1}{6} lim_{x \to 0} \frac{0 - (0 - \frac{d}{d x} [x])}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.5

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

$\frac{1}{6} lim_{x \to 0} \frac{0 - (0 - 1 \cdot 1)}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.6

Multiply $- 1$ by $1$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 - (0 - 1)}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.7

Subtract $1$ from $0$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 - - 1}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.4.8

Multiply $- 1$ by $- 1$ .

$\frac{1}{6} lim_{x \to 0} \frac{0 + 1}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.5

Add $0$ and $1$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{\frac{d}{d x} [(6 - x) x]}$

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 6 - x$ and $g (x) = x$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{(6 - x) \frac{d}{d x} [x] + x \frac{d}{d x} [6 - x]}$

Step 3.3.7

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

$\frac{1}{6} lim_{x \to 0} \frac{1}{(6 - x) \cdot 1 + x \frac{d}{d x} [6 - x]}$

Step 3.3.8

Multiply $6 - x$ by $1$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x \frac{d}{d x} [6 - x]}$

Step 3.3.9

By the Sum Rule, the derivative of $6 - x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x]$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x (\frac{d}{d x} [6] + \frac{d}{d x} [- x])}$

Step 3.3.10

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x (0 + \frac{d}{d x} [- x])}$

Step 3.3.11

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x \frac{d}{d x} [- x]}$

Step 3.3.12

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

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x \cdot - 1 \frac{d}{d x} [x]}$

Step 3.3.13

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

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x \cdot - 1 \cdot 1}$

Step 3.3.14

Multiply $- 1$ by $1$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x + x \cdot - 1}$

Step 3.3.15

Move $- 1$ to the left of $x$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x - 1 \cdot x}$

Step 3.3.16

Rewrite $- 1 x$ as $- x$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - x - x}$

Step 3.3.17

Subtract $x$ from $- x$ .

$\frac{1}{6} lim_{x \to 0} \frac{1}{6 - 2 x}$

Step 3.3.18

Reorder terms.

$\frac{1}{6} lim_{x \to 0} \frac{1}{- 2 x + 6}$

Step 4

Evaluate the limit.

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Step 4.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{lim_{x \to 0} 1}{lim_{x \to 0} - 2 x + 6}$

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{1}{lim_{x \to 0} - 2 x + 6}$

Step 4.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{1}{- lim_{x \to 0} 2 x + lim_{x \to 0} 6}$

Step 4.4

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

$\frac{1}{6} \cdot \frac{1}{- 2 lim_{x \to 0} x + lim_{x \to 0} 6}$

Step 4.5

Evaluate the limit of $6$ which is constant as $x$ approaches $0$ .

$\frac{1}{6} \cdot \frac{1}{- 2 lim_{x \to 0} x + 6}$

Step 5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{6} \cdot \frac{1}{- 2 \cdot 0 + 6}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Multiply $- 2$ by $0$ .

$\frac{1}{6} \cdot \frac{1}{0 + 6}$

Step 6.1.2

Add $0$ and $6$ .

$\frac{1}{6} \cdot \frac{1}{6}$

Step 6.2

Multiply $\frac{1}{6} \cdot \frac{1}{6}$ .

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Step 6.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{6}$ .

$\frac{1}{6 \cdot 6}$

Step 6.2.2

Multiply $6$ by $6$ .

$\frac{1}{36}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{36}$

Decimal Form:

$0.02\overline{7}$