Calculus Examples

$lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{x^{2} - \frac{1}{64}}$

Step 1

Combine terms.

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Step 1.1

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{x^{2} \cdot \frac{64}{64} - \frac{1}{64}}$

Step 1.2

Combine $x^{2}$ and $\frac{64}{64}$ .

$lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{\frac{x^{2} \cdot 64}{64} - \frac{1}{64}}$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{\frac{x^{2} \cdot 64 - 1}{64}}$

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.

$lim_{x \to - \frac{1}{8}} (x + 8 x^{2}) \frac{64}{x^{2} \cdot 64 - 1}$

Step 2.1.2

Multiply $x + 8 x^{2}$ by $\frac{64}{x^{2} \cdot 64 - 1}$ .

$lim_{x \to - \frac{1}{8}} \frac{(x + 8 x^{2}) \cdot 64}{x^{2} \cdot 64 - 1}$

Step 2.2

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$64 lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{x^{2} \cdot 64 - 1}$

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.

$64 \frac{lim_{x \to - \frac{1}{8}} x + 8 x^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

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 $- \frac{1}{8}$ .

$64 \frac{lim_{x \to - \frac{1}{8}} x + lim_{x \to - \frac{1}{8}} 8 x^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.2

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$64 \frac{lim_{x \to - \frac{1}{8}} x + 8 lim_{x \to - \frac{1}{8}} x^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.3

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

$64 \frac{lim_{x \to - \frac{1}{8}} x + 8 {(lim_{x \to - \frac{1}{8}} x)}^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.4

Evaluate the limits by plugging in $- \frac{1}{8}$ for all occurrences of $x$ .

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Step 3.1.2.4.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$64 \frac{- \frac{1}{8} + 8 {(lim_{x \to - \frac{1}{8}} x)}^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.4.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$64 \frac{- \frac{1}{8} + 8 {(- \frac{1}{8})}^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5

Simplify the answer.

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Step 3.1.2.5.1

Simplify each term.

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Step 3.1.2.5.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.2.5.1.1.1

Apply the product rule to $- \frac{1}{8}$ .

$64 \frac{- \frac{1}{8} + 8 {(- 1)}^{2} {(\frac{1}{8})}^{2}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.1.2

Apply the product rule to $\frac{1}{8}$ .

$64 \frac{- \frac{1}{8} + 8 {(- 1)}^{2} \frac{1^{2}}{8^{2}}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.2

Raise $- 1$ to the power of $2$ .

$64 \frac{- \frac{1}{8} + 8 \cdot 1 \frac{1^{2}}{8^{2}}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.3

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$64 \frac{- \frac{1}{8} + 8 \frac{1^{2}}{8^{2}}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.4

Cancel the common factor of $8$ .

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Step 3.1.2.5.1.4.1

Factor $8$ out of $8^{2}$ .

$64 \frac{- \frac{1}{8} + 8 \frac{1^{2}}{8 \cdot 8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.4.2

Cancel the common factor.

$64 \frac{- \frac{1}{8} + 8 \frac{1^{2}}{8 \cdot 8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.4.3

Rewrite the expression.

$64 \frac{- \frac{1}{8} + \frac{1^{2}}{8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.1.5

One to any power is one.

$64 \frac{- \frac{1}{8} + \frac{1}{8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.2

Combine the numerators over the common denominator.

$64 \frac{\frac{- 1 + 1}{8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.3

Add $- 1$ and $1$ .

$64 \frac{\frac{0}{8}}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.2.5.4

Divide $0$ by $8$ .

$64 \frac{0}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - 1}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{8}$ .

$64 \frac{0}{lim_{x \to - \frac{1}{8}} x^{2} \cdot 64 - lim_{x \to - \frac{1}{8}} 1}$

Step 3.1.3.1.2

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$64 \frac{0}{64 lim_{x \to - \frac{1}{8}} x^{2} - lim_{x \to - \frac{1}{8}} 1}$

Step 3.1.3.1.3

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

$64 \frac{0}{64 {(lim_{x \to - \frac{1}{8}} x)}^{2} - lim_{x \to - \frac{1}{8}} 1}$

Step 3.1.3.1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{8}$ .

$64 \frac{0}{64 {(lim_{x \to - \frac{1}{8}} x)}^{2} - 1 \cdot 1}$

Step 3.1.3.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$64 \frac{0}{64 {(- \frac{1}{8})}^{2} - 1 \cdot 1}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Simplify each term.

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Step 3.1.3.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.3.3.1.1.1

Apply the product rule to $- \frac{1}{8}$ .

$64 \frac{0}{64 {(- 1)}^{2} {(\frac{1}{8})}^{2} - 1 \cdot 1}$

Step 3.1.3.3.1.1.2

Apply the product rule to $\frac{1}{8}$ .

$64 \frac{0}{64 {(- 1)}^{2} \frac{1^{2}}{8^{2}} - 1 \cdot 1}$

Step 3.1.3.3.1.2

Raise $- 1$ to the power of $2$ .

$64 \frac{0}{64 \cdot 1 \frac{1^{2}}{8^{2}} - 1 \cdot 1}$

Step 3.1.3.3.1.3

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

$64 \frac{0}{64 \frac{1^{2}}{8^{2}} - 1 \cdot 1}$

Step 3.1.3.3.1.4

One to any power is one.

$64 \frac{0}{64 \frac{1}{8^{2}} - 1 \cdot 1}$

Step 3.1.3.3.1.5

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

$64 \frac{0}{64 (\frac{1}{64}) - 1 \cdot 1}$

Step 3.1.3.3.1.6

Cancel the common factor of $64$ .

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Step 3.1.3.3.1.6.1

Cancel the common factor.

$64 \frac{0}{64 (\frac{1}{64}) - 1 \cdot 1}$

Step 3.1.3.3.1.6.2

Rewrite the expression.

$64 \frac{0}{1 - 1 \cdot 1}$

Step 3.1.3.3.1.7

Multiply $- 1$ by $1$ .

$64 \frac{0}{1 - 1}$

Step 3.1.3.3.2

Subtract $1$ from $1$ .

$64 (\frac{0}{0})$

Step 3.1.3.3.3

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

Undefined

$64 (\frac{0}{0})$

Step 3.1.3.4

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

Undefined

$64 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$64 (\frac{0}{0})$

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.

$lim_{x \to - \frac{1}{8}} \frac{x + 8 x^{2}}{x^{2} \cdot 64 - 1} = lim_{x \to - \frac{1}{8}} \frac{\frac{d}{d x} [x + 8 x^{2}]}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$64 lim_{x \to - \frac{1}{8}} \frac{\frac{d}{d x} [x + 8 x^{2}]}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.2

By the Sum Rule, the derivative of $x + 8 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8 x^{2}]$ .

$64 lim_{x \to - \frac{1}{8}} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [8 x^{2}]}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.3

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

$64 lim_{x \to - \frac{1}{8}} \frac{1 + \frac{d}{d x} [8 x^{2}]}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

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Step 3.3.4.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$64 lim_{x \to - \frac{1}{8}} \frac{1 + 8 \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

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 $n = 2$ .

$64 lim_{x \to - \frac{1}{8}} \frac{1 + 8 \cdot 2 x}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.4.3

Multiply $2$ by $8$ .

$64 lim_{x \to - \frac{1}{8}} \frac{1 + 16 x}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.5

Reorder terms.

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{\frac{d}{d x} [x^{2} \cdot 64 - 1]}$

Step 3.3.6

By the Sum Rule, the derivative of $x^{2} \cdot 64 - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2} \cdot 64] + \frac{d}{d x} [- 1]$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{\frac{d}{d x} [x^{2} \cdot 64] + \frac{d}{d x} [- 1]}$

Step 3.3.7

Evaluate $\frac{d}{d x} [x^{2} \cdot 64]$ .

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Step 3.3.7.1

Move $64$ to the left of $x^{2}$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{\frac{d}{d x} [64 \cdot x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.7.2

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]}$

Step 3.3.7.3

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

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{64 \cdot 2 x + \frac{d}{d x} [- 1]}$

Step 3.3.7.4

Multiply $2$ by $64$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{128 x + \frac{d}{d x} [- 1]}$

Step 3.3.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{128 x + 0}$

Step 3.3.9

Add $128 x$ and $0$ .

$64 lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{128 x}$

Step 4

Evaluate the limit.

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Step 4.1

Move the term $\frac{1}{128}$ outside of the limit because it is constant with respect to $x$ .

$64 (\frac{1}{128}) lim_{x \to - \frac{1}{8}} \frac{16 x + 1}{x}$

Step 4.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- \frac{1}{8}$ .

$64 (\frac{1}{128}) \frac{lim_{x \to - \frac{1}{8}} 16 x + 1}{lim_{x \to - \frac{1}{8}} x}$

Step 4.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \frac{1}{8}$ .

$64 (\frac{1}{128}) \frac{lim_{x \to - \frac{1}{8}} 16 x + lim_{x \to - \frac{1}{8}} 1}{lim_{x \to - \frac{1}{8}} x}$

Step 4.4

Move the term $16$ outside of the limit because it is constant with respect to $x$ .

$64 (\frac{1}{128}) \frac{16 lim_{x \to - \frac{1}{8}} x + lim_{x \to - \frac{1}{8}} 1}{lim_{x \to - \frac{1}{8}} x}$

Step 4.5

Evaluate the limit of $1$ which is constant as $x$ approaches $- \frac{1}{8}$ .

$64 (\frac{1}{128}) \frac{16 lim_{x \to - \frac{1}{8}} x + 1}{lim_{x \to - \frac{1}{8}} x}$

Step 5

Evaluate the limits by plugging in $- \frac{1}{8}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$64 (\frac{1}{128}) \frac{16 (- \frac{1}{8}) + 1}{lim_{x \to - \frac{1}{8}} x}$

Step 5.2

Evaluate the limit of $x$ by plugging in $- \frac{1}{8}$ for $x$ .

$64 (\frac{1}{128}) \frac{16 (- \frac{1}{8}) + 1}{- \frac{1}{8}}$

Step 6

Simplify the answer.

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Step 6.1

Cancel the common factor of $64$ .

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Step 6.1.1

Factor $64$ out of $128$ .

$64 \frac{1}{64 (2)} \frac{16 (- \frac{1}{8}) + 1}{- \frac{1}{8}}$

Step 6.1.2

Cancel the common factor.

$64 \frac{1}{64 \cdot 2} \frac{16 (- \frac{1}{8}) + 1}{- \frac{1}{8}}$

Step 6.1.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{16 (- \frac{1}{8}) + 1}{- \frac{1}{8}}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} ((16 (- \frac{1}{8}) + 1) (- 1 \cdot 8))$

Step 6.3

Simplify each term.

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Step 6.3.1

Cancel the common factor of $8$ .

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Step 6.3.1.1

Move the leading negative in $- \frac{1}{8}$ into the numerator.

$\frac{1}{2} ((16 (\frac{- 1}{8}) + 1) (- 1 \cdot 8))$

Step 6.3.1.2

Factor $8$ out of $16$ .

$\frac{1}{2} ((8 (2) \frac{- 1}{8} + 1) (- 1 \cdot 8))$

Step 6.3.1.3

Cancel the common factor.

$\frac{1}{2} ((8 \cdot 2 \frac{- 1}{8} + 1) (- 1 \cdot 8))$

Step 6.3.1.4

Rewrite the expression.

$\frac{1}{2} ((2 \cdot - 1 + 1) (- 1 \cdot 8))$

Step 6.3.2

Multiply $2$ by $- 1$ .

$\frac{1}{2} ((- 2 + 1) (- 1 \cdot 8))$

Step 6.4

Add $- 2$ and $1$ .

$\frac{1}{2} (- 1 (- 1 \cdot 8))$

Step 6.5

Multiply $- 1 (- 1 \cdot 8)$ .

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Step 6.5.1

Multiply $- 1$ by $8$ .

$\frac{1}{2} (- 1 \cdot - 8)$

Step 6.5.2

Multiply $- 1$ by $- 8$ .

$\frac{1}{2} \cdot 8$

Step 6.6

Cancel the common factor of $2$ .

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Step 6.6.1

Factor $2$ out of $8$ .

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

Step 6.6.2

Cancel the common factor.

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

Step 6.6.3

Rewrite the expression.

$4$