Calculus Examples

$lim_{x \to \frac{1}{2}} \frac{2 - 4 x}{8 - 64 x^{3}}$

Step 1

Cancel the common factor of $2 - 4 x$ and $8 - 64 x^{3}$ .

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Step 1.1

Factor $2$ out of $2$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1) - 4 x}{8 - 64 x^{3}}$

Step 1.2

Factor $2$ out of $- 4 x$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1) + 2 (- 2 x)}{8 - 64 x^{3}}$

Step 1.3

Factor $2$ out of $2 (1) + 2 (- 2 x)$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1 - 2 x)}{8 - 64 x^{3}}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $8$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1 - 2 x)}{2 \cdot 4 - 64 x^{3}}$

Step 1.4.2

Factor $2$ out of $- 64 x^{3}$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1 - 2 x)}{2 \cdot 4 + 2 (- 32 x^{3})}$

Step 1.4.3

Factor $2$ out of $2 (4) + 2 (- 32 x^{3})$ .

$lim_{x \to \frac{1}{2}} \frac{2 (1 - 2 x)}{2 (4 - 32 x^{3})}$

Step 1.4.4

Cancel the common factor.

$lim_{x \to \frac{1}{2}} \frac{2 (1 - 2 x)}{2 (4 - 32 x^{3})}$

Step 1.4.5

Rewrite the expression.

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

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

Step 2.1.2.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{2}$ .

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

Step 2.1.2.1.3

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

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

Step 2.1.2.2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{1 - 2 (\frac{1}{2})}{lim_{x \to \frac{1}{2}} 4 - 32 x^{3}}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Cancel the common factor of $2$ .

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Step 2.1.2.3.1.1

Factor $2$ out of $- 2$ .

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

Step 2.1.2.3.1.2

Cancel the common factor.

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

Step 2.1.2.3.1.3

Rewrite the expression.

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

Step 2.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to \frac{1}{2}} 4 - 32 x^{3}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{0}{lim_{x \to \frac{1}{2}} 4 - lim_{x \to \frac{1}{2}} 32 x^{3}}$

Step 2.1.3.1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{0}{4 - lim_{x \to \frac{1}{2}} 32 x^{3}}$

Step 2.1.3.1.3

Move the term $32$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{4 - 32 lim_{x \to \frac{1}{2}} x^{3}}$

Step 2.1.3.1.4

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{4 - 32 {(lim_{x \to \frac{1}{2}} x)}^{3}}$

Step 2.1.3.2

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{0}{4 - 32 {(\frac{1}{2})}^{3}}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Simplify each term.

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Step 2.1.3.3.1.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{0}{4 - 32 \frac{1^{3}}{2^{3}}}$

Step 2.1.3.3.1.2

One to any power is one.

$\frac{0}{4 - 32 \frac{1}{2^{3}}}$

Step 2.1.3.3.1.3

Raise $2$ to the power of $3$ .

$\frac{0}{4 - 32 (\frac{1}{8})}$

Step 2.1.3.3.1.4

Cancel the common factor of $8$ .

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Step 2.1.3.3.1.4.1

Factor $8$ out of $- 32$ .

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

Step 2.1.3.3.1.4.2

Cancel the common factor.

$\frac{0}{4 + 8 \cdot - 4 \frac{1}{8}}$

Step 2.1.3.3.1.4.3

Rewrite the expression.

$\frac{0}{4 - 4}$

Step 2.1.3.3.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 2.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{x \to \frac{1}{2}} \frac{1 - 2 x}{4 - 32 x^{3}} = lim_{x \to \frac{1}{2}} \frac{\frac{d}{d x} [1 - 2 x]}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{1}{2}} \frac{\frac{d}{d x} [1 - 2 x]}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.2

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

$lim_{x \to \frac{1}{2}} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x]}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.3

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

$lim_{x \to \frac{1}{2}} \frac{0 + \frac{d}{d x} [- 2 x]}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.4

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

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Step 2.3.4.1

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

$lim_{x \to \frac{1}{2}} \frac{0 - 2 \frac{d}{d x} [x]}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.4.2

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

$lim_{x \to \frac{1}{2}} \frac{0 - 2 \cdot 1}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.4.3

Multiply $- 2$ by $1$ .

$lim_{x \to \frac{1}{2}} \frac{0 - 2}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.5

Subtract $2$ from $0$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{\frac{d}{d x} [4 - 32 x^{3}]}$

Step 2.3.6

By the Sum Rule, the derivative of $4 - 32 x^{3}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 32 x^{3}]$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{\frac{d}{d x} [4] + \frac{d}{d x} [- 32 x^{3}]}$

Step 2.3.7

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

$lim_{x \to \frac{1}{2}} \frac{- 2}{0 + \frac{d}{d x} [- 32 x^{3}]}$

Step 2.3.8

Evaluate $\frac{d}{d x} [- 32 x^{3}]$ .

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Step 2.3.8.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- 32 x^{3}$ with respect to $x$ is $- 32 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{0 - 32 \frac{d}{d x} [x^{3}]}$

Step 2.3.8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{0 - 32 (3 x^{2})}$

Step 2.3.8.3

Multiply $3$ by $- 32$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{0 - 96 x^{2}}$

Step 2.3.9

Subtract $96 x^{2}$ from $0$ .

$lim_{x \to \frac{1}{2}} \frac{- 2}{- 96 x^{2}}$

Step 2.4

Cancel the common factor of $- 2$ and $- 96$ .

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Step 2.4.1

Factor $- 2$ out of $- 2$ .

$lim_{x \to \frac{1}{2}} \frac{- 2 (1)}{- 96 x^{2}}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $- 2$ out of $- 96 x^{2}$ .

$lim_{x \to \frac{1}{2}} \frac{- 2 (1)}{- 2 (48 x^{2})}$

Step 2.4.2.2

Cancel the common factor.

$lim_{x \to \frac{1}{2}} \frac{- 2 \cdot 1}{- 2 (48 x^{2})}$

Step 2.4.2.3

Rewrite the expression.

$lim_{x \to \frac{1}{2}} \frac{1}{48 x^{2}}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $\frac{1}{48}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{48} lim_{x \to \frac{1}{2}} \frac{1}{x^{2}}$

Step 3.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{1}{2}$ .

$\frac{1}{48} \cdot \frac{lim_{x \to \frac{1}{2}} 1}{lim_{x \to \frac{1}{2}} x^{2}}$

Step 3.3

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{1}{2}$ .

$\frac{1}{48} \cdot \frac{1}{lim_{x \to \frac{1}{2}} x^{2}}$

Step 3.4

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

$\frac{1}{48} \cdot \frac{1}{{(lim_{x \to \frac{1}{2}} x)}^{2}}$

Step 4

Evaluate the limit of $x$ by plugging in $\frac{1}{2}$ for $x$ .

$\frac{1}{48} \cdot \frac{1}{{(\frac{1}{2})}^{2}}$

Step 5

Simplify the answer.

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Step 5.1

Combine.

$\frac{1 \cdot 1}{48 {(\frac{1}{2})}^{2}}$

Step 5.2

Multiply $1$ by $1$ .

$\frac{1}{48 {(\frac{1}{2})}^{2}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1}{48 \frac{1^{2}}{2^{2}}}$

Step 5.3.2

One to any power is one.

$\frac{1}{48 \frac{1}{2^{2}}}$

Step 5.3.3

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

$\frac{1}{48 (\frac{1}{4})}$

Step 5.4

Combine $48$ and $\frac{1}{4}$ .

$\frac{1}{\frac{48}{4}}$

Step 5.5

Divide $48$ by $4$ .

$\frac{1}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{12}$

Decimal Form:

$0.08\overline{3}$