Calculus Examples

$lim_{x \to 8} \frac{\frac{2}{x} - \frac{1}{x - 4}}{x - 8}$

Step 1

Combine terms.

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Step 1.1

To write $\frac{2}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$lim_{x \to 8} \frac{\frac{2}{x} \cdot \frac{x - 4}{x - 4} - \frac{1}{x - 4}}{x - 8}$

Step 1.2

To write $- \frac{1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 8} \frac{\frac{2}{x} \cdot \frac{x - 4}{x - 4} - \frac{1}{x - 4} \cdot \frac{x}{x}}{x - 8}$

Step 1.3

Write each expression with a common denominator of $x (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2}{x}$ by $\frac{x - 4}{x - 4}$ .

$lim_{x \to 8} \frac{\frac{2 (x - 4)}{x (x - 4)} - \frac{1}{x - 4} \cdot \frac{x}{x}}{x - 8}$

Step 1.3.2

Multiply $\frac{1}{x - 4}$ by $\frac{x}{x}$ .

$lim_{x \to 8} \frac{\frac{2 (x - 4)}{x (x - 4)} - \frac{x}{(x - 4) x}}{x - 8}$

Step 1.3.3

Reorder the factors of $(x - 4) x$ .

$lim_{x \to 8} \frac{\frac{2 (x - 4)}{x (x - 4)} - \frac{x}{x (x - 4)}}{x - 8}$

Step 1.4

Combine the numerators over the common denominator.

$lim_{x \to 8} \frac{\frac{2 (x - 4) - x}{x (x - 4)}}{x - 8}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 8} \frac{2 (x - 4) - x}{x (x - 4)} \cdot \frac{1}{x - 8}$

Step 2.2

Multiply $\frac{2 (x - 4) - x}{x (x - 4)}$ by $\frac{1}{x - 8}$ .

$lim_{x \to 8} \frac{2 (x - 4) - x}{x (x - 4) (x - 8)}$

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.

$\frac{lim_{x \to 8} 2 (x - 4) - x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 2 (x - 4) - lim_{x \to 8} x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.2

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

$\frac{2 lim_{x \to 8} x - 4 - lim_{x \to 8} x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{2 (lim_{x \to 8} x - lim_{x \to 8} 4) - lim_{x \to 8} x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.4

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{2 (lim_{x \to 8} x - 1 \cdot 4) - lim_{x \to 8} x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.5

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 3.1.2.5.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 (8 - 1 \cdot 4) - lim_{x \to 8} x}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.5.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2 (8 - 1 \cdot 4) - 8}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.6

Simplify the answer.

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Step 3.1.2.6.1

Simplify each term.

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Step 3.1.2.6.1.1

Multiply $- 1$ by $4$ .

$\frac{2 (8 - 4) - 8}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.6.1.2

Subtract $4$ from $8$ .

$\frac{2 \cdot 4 - 8}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.6.1.3

Multiply $2$ by $4$ .

$\frac{8 - 8}{lim_{x \to 8} x (x - 4) (x - 8)}$

Step 3.1.2.6.2

Subtract $8$ from $8$ .

$\frac{0}{lim_{x \to 8} x (x - 4) (x - 8)}$

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 $8$ .

$\frac{0}{lim_{x \to 8} x \cdot lim_{x \to 8} x - 4 \cdot lim_{x \to 8} x - 8}$

Step 3.1.3.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{0}{lim_{x \to 8} x \cdot (lim_{x \to 8} x - lim_{x \to 8} 4) \cdot lim_{x \to 8} x - 8}$

Step 3.1.3.3

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$\frac{0}{lim_{x \to 8} x \cdot (lim_{x \to 8} x - 1 \cdot 4) \cdot lim_{x \to 8} x - 8}$

Step 3.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{0}{lim_{x \to 8} x \cdot (lim_{x \to 8} x - 1 \cdot 4) \cdot (lim_{x \to 8} x - lim_{x \to 8} 8)}$

Step 3.1.3.5

Evaluate the limit of $8$ which is constant as $x$ approaches $8$ .

$\frac{0}{lim_{x \to 8} x \cdot (lim_{x \to 8} x - 1 \cdot 4) \cdot (lim_{x \to 8} x - 1 \cdot 8)}$

Step 3.1.3.6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 3.1.3.6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{0}{8 \cdot (lim_{x \to 8} x - 1 \cdot 4) \cdot (lim_{x \to 8} x - 1 \cdot 8)}$

Step 3.1.3.6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{0}{8 \cdot (8 - 1 \cdot 4) \cdot (lim_{x \to 8} x - 1 \cdot 8)}$

Step 3.1.3.6.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{0}{8 \cdot (8 - 1 \cdot 4) \cdot (8 - 1 \cdot 8)}$

Step 3.1.3.7

Simplify the answer.

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Step 3.1.3.7.1

Multiply $- 1$ by $4$ .

$\frac{0}{8 \cdot (8 - 4) \cdot (8 - 1 \cdot 8)}$

Step 3.1.3.7.2

Subtract $4$ from $8$ .

$\frac{0}{8 \cdot 4 \cdot (8 - 1 \cdot 8)}$

Step 3.1.3.7.3

Multiply $8$ by $4$ .

$\frac{0}{32 \cdot (8 - 1 \cdot 8)}$

Step 3.1.3.7.4

Multiply $- 1$ by $8$ .

$\frac{0}{32 \cdot (8 - 8)}$

Step 3.1.3.7.5

Subtract $8$ from $8$ .

$\frac{0}{32 \cdot 0}$

Step 3.1.3.7.6

Multiply $32$ by $0$ .

$\frac{0}{0}$

Step 3.1.3.7.7

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

Undefined

$\frac{0}{0}$

Step 3.1.3.8

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

Undefined

$\frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 8} \frac{2 (x - 4) - x}{x (x - 4) (x - 8)} = lim_{x \to 8} \frac{\frac{d}{d x} [2 (x - 4) - x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 8} \frac{\frac{d}{d x} [2 (x - 4) - x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.2

By the Sum Rule, the derivative of $2 (x - 4) - x$ with respect to $x$ is $\frac{d}{d x} [2 (x - 4)] + \frac{d}{d x} [- x]$ .

$lim_{x \to 8} \frac{\frac{d}{d x} [2 (x - 4)] + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3

Evaluate $\frac{d}{d x} [2 (x - 4)]$ .

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Step 3.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 (x - 4)$ with respect to $x$ is $2 \frac{d}{d x} [x - 4]$ .

$lim_{x \to 8} \frac{2 \frac{d}{d x} [x - 4] + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3.2

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 8} \frac{2 (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3.3

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

$lim_{x \to 8} \frac{2 (1 + \frac{d}{d x} [- 4]) + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 8} \frac{2 (1 + 0) + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3.5

Add $1$ and $0$ .

$lim_{x \to 8} \frac{2 \cdot 1 + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.3.6

Multiply $2$ by $1$ .

$lim_{x \to 8} \frac{2 + \frac{d}{d x} [- x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.4

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

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Step 3.3.4.1

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

$lim_{x \to 8} \frac{2 - \frac{d}{d x} [x]}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.4.2

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

$lim_{x \to 8} \frac{2 - 1 \cdot 1}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.4.3

Multiply $- 1$ by $1$ .

$lim_{x \to 8} \frac{2 - 1}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

Step 3.3.5

Subtract $1$ from $2$ .

$lim_{x \to 8} \frac{1}{\frac{d}{d x} [x (x - 4) (x - 8)]}$

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 $f (x) = x (x - 4)$ and $g (x) = x - 8$ .

$lim_{x \to 8} \frac{1}{x (x - 4) \frac{d}{d x} [x - 8] + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.7

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$lim_{x \to 8} \frac{1}{x (x - 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.8

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

$lim_{x \to 8} \frac{1}{x (x - 4) (1 + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.9

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to 8} \frac{1}{x (x - 4) (1 + 0) + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.10

Add $1$ and $0$ .

$lim_{x \to 8} \frac{1}{x (x - 4) \cdot 1 + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.11

Multiply $x$ by $1$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) \frac{d}{d x} [x (x - 4)]}$

Step 3.3.12

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 $f (x) = x$ and $g (x) = x - 4$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x \frac{d}{d x} [x - 4] + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.13

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x (\frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.14

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

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x (1 + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.15

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x (1 + 0) + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.16

Add $1$ and $0$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x \cdot 1 + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.17

Multiply $x$ by $1$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x + (x - 4) \frac{d}{d x} [x])}$

Step 3.3.18

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

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x + (x - 4) \cdot 1)}$

Step 3.3.19

Multiply $x - 4$ by $1$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (x + x - 4)}$

Step 3.3.20

Add $x$ and $x$ .

$lim_{x \to 8} \frac{1}{x (x - 4) + (x - 8) (2 x - 4)}$

Step 3.3.21

Simplify.

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Step 3.3.21.1

Apply the distributive property.

$lim_{x \to 8} \frac{1}{x \cdot x + x \cdot - 4 + (x - 8) (2 x - 4)}$

Step 3.3.21.2

Apply the distributive property.

$lim_{x \to 8} \frac{1}{x \cdot x + x \cdot - 4 + (x - 8) (2 x) + (x - 8) \cdot - 4}$

Step 3.3.21.3

Apply the distributive property.

$lim_{x \to 8} \frac{1}{x \cdot x + x \cdot - 4 + x (2 x) - 8 (2 x) + (x - 8) \cdot - 4}$

Step 3.3.21.4

Apply the distributive property.

$lim_{x \to 8} \frac{1}{x \cdot x + x \cdot - 4 + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5

Combine terms.

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Step 3.3.21.5.1

Raise $x$ to the power of $1$ .

$lim_{x \to 8} \frac{1}{x^{1} x + x \cdot - 4 + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.2

Raise $x$ to the power of $1$ .

$lim_{x \to 8} \frac{1}{x^{1} x^{1} + x \cdot - 4 + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 8} \frac{1}{x^{1 + 1} + x \cdot - 4 + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.4

Add $1$ and $1$ .

$lim_{x \to 8} \frac{1}{x^{2} + x \cdot - 4 + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.5

Move $- 4$ to the left of $x$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 \cdot x + x (2 x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.6

Raise $x$ to the power of $1$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 (x^{1} x) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.7

Raise $x$ to the power of $1$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 (x^{1} x^{1}) - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{1 + 1} - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.9

Add $1$ and $1$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{2} - 8 (2 x) + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.10

Multiply $2$ by $- 8$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{2} - 16 x + x \cdot - 4 - 8 \cdot - 4}$

Step 3.3.21.5.11

Move $- 4$ to the left of $x$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{2} - 16 x - 4 \cdot x - 8 \cdot - 4}$

Step 3.3.21.5.12

Multiply $- 8$ by $- 4$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{2} - 16 x - 4 x + 32}$

Step 3.3.21.5.13

Subtract $4 x$ from $- 16 x$ .

$lim_{x \to 8} \frac{1}{x^{2} - 4 x + 2 x^{2} - 20 x + 32}$

Step 3.3.21.5.14

Add $x^{2}$ and $2 x^{2}$ .

$lim_{x \to 8} \frac{1}{3 x^{2} - 4 x - 20 x + 32}$

Step 3.3.21.5.15

Subtract $20 x$ from $- 4 x$ .

$lim_{x \to 8} \frac{1}{3 x^{2} - 24 x + 32}$

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 $8$ .

$\frac{lim_{x \to 8} 1}{lim_{x \to 8} 3 x^{2} - 24 x + 32}$

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{1}{lim_{x \to 8} 3 x^{2} - 24 x + 32}$

Step 4.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{1}{lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 24 x + lim_{x \to 8} 32}$

Step 4.4

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

$\frac{1}{3 lim_{x \to 8} x^{2} - lim_{x \to 8} 24 x + lim_{x \to 8} 32}$

Step 4.5

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

$\frac{1}{3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 24 x + lim_{x \to 8} 32}$

Step 4.6

Move the term $24$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3 {(lim_{x \to 8} x)}^{2} - 24 lim_{x \to 8} x + lim_{x \to 8} 32}$

Step 4.7

Evaluate the limit of $32$ which is constant as $x$ approaches $8$ .

$\frac{1}{3 {(lim_{x \to 8} x)}^{2} - 24 lim_{x \to 8} x + 32}$

Step 5

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{3 \cdot 8^{2} - 24 lim_{x \to 8} x + 32}$

Step 5.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{3 \cdot 8^{2} - 24 \cdot 8 + 32}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $8$ to the power of $2$ .

$\frac{1}{3 \cdot 64 - 24 \cdot 8 + 32}$

Step 6.2

Multiply $3$ by $64$ .

$\frac{1}{192 - 24 \cdot 8 + 32}$

Step 6.3

Multiply $- 24$ by $8$ .

$\frac{1}{192 - 192 + 32}$

Step 6.4

Subtract $192$ from $192$ .

$\frac{1}{0 + 32}$

Step 6.5

Add $0$ and $32$ .

$\frac{1}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{32}$

Decimal Form:

$0.03125$