Calculus Examples

$lim_{x \to 4} \frac{x^{3} \sqrt{x} - 128}{x - 4}$

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.

$\frac{lim_{x \to 4} x^{3} \sqrt{x} - 128}{lim_{x \to 4} x - 4}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x^{3} \sqrt{x} - lim_{x \to 4} 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x^{3} \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.3

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

$\frac{{(lim_{x \to 4} x)}^{3} \cdot lim_{x \to 4} \sqrt{x} - lim_{x \to 4} 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.4

Move the limit under the radical sign.

$\frac{{(lim_{x \to 4} x)}^{3} \cdot \sqrt{lim_{x \to 4} x} - lim_{x \to 4} 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.5

Evaluate the limit of $128$ which is constant as $x$ approaches $4$ .

$\frac{{(lim_{x \to 4} x)}^{3} \cdot \sqrt{lim_{x \to 4} x} - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.6

Evaluate the limits by plugging in $4$ for all occurrences of $x$ .

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Step 1.1.2.6.1

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4^{3} \cdot \sqrt{lim_{x \to 4} x} - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.6.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4^{3} \cdot \sqrt{4} - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7

Simplify the answer.

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Step 1.1.2.7.1

Simplify each term.

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Step 1.1.2.7.1.1

Raise $4$ to the power of $3$ .

$\frac{64 \cdot \sqrt{4} - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{64 \cdot \sqrt{2^{2}} - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{64 \cdot 2 - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7.1.4

Multiply $64$ by $2$ .

$\frac{128 - 1 \cdot 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7.1.5

Multiply $- 1$ by $128$ .

$\frac{128 - 128}{lim_{x \to 4} x - 4}$

Step 1.1.2.7.2

Subtract $128$ from $128$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $4$ .

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

Step 1.1.3.3.2

Subtract $4$ from $4$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

$lim_{x \to 4} \frac{x^{3} \sqrt{x} - 128}{x - 4} = lim_{x \to 4} \frac{\frac{d}{d x} [x^{3} \sqrt{x} - 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{3} \sqrt{x} - 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.2

By the Sum Rule, the derivative of $x^{3} \sqrt{x} - 128$ with respect to $x$ is $\frac{d}{d x} [x^{3} \sqrt{x}] + \frac{d}{d x} [- 128]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{3} \sqrt{x}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3

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

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{3} x^{\frac{1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2

Multiply $x^{3}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.3.3.2.1

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

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{3 + \frac{1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{3 \cdot \frac{2}{2} + \frac{1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2.3

Combine $3$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{\frac{3 \cdot 2}{2} + \frac{1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2.4

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{\frac{3 \cdot 2 + 1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2.5

Simplify the numerator.

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Step 1.3.3.2.5.1

Multiply $3$ by $2$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{\frac{6 + 1}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.2.5.2

Add $6$ and $1$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x^{\frac{7}{2}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{2}$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{7}{2} - 1} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.5

Combine $- 1$ and $\frac{2}{2}$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.6

Combine the numerators over the common denominator.

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.7

Simplify the numerator.

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Step 1.3.3.7.1

Multiply $- 1$ by $2$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{7 - 2}{2}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.3.7.2

Subtract $2$ from $7$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{5}{2}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 4]}$

Step 1.3.4

Since $- 128$ is constant with respect to $x$ , the derivative of $- 128$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{5}{2}} + 0}{\frac{d}{d x} [x - 4]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $\frac{7}{2} x^{\frac{5}{2}}$ and $0$ .

$lim_{x \to 4} \frac{\frac{7}{2} x^{\frac{5}{2}}}{\frac{d}{d x} [x - 4]}$

Step 1.3.5.2

Combine $\frac{7}{2}$ and $x^{\frac{5}{2}}$ .

$lim_{x \to 4} \frac{\frac{7 x^{\frac{5}{2}}}{2}}{\frac{d}{d x} [x - 4]}$

Step 1.3.6

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

$lim_{x \to 4} \frac{\frac{7 x^{\frac{5}{2}}}{2}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 4]}$

Step 1.3.7

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

$lim_{x \to 4} \frac{\frac{7 x^{\frac{5}{2}}}{2}}{1 + \frac{d}{d x} [- 4]}$

Step 1.3.8

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

$lim_{x \to 4} \frac{\frac{7 x^{\frac{5}{2}}}{2}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 4} \frac{\frac{7 x^{\frac{5}{2}}}{2}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 4} \frac{7 x^{\frac{5}{2}}}{2} \cdot 1$

Step 1.5

Multiply $\frac{7 x^{\frac{5}{2}}}{2}$ by $1$ .

$lim_{x \to 4} \frac{7 x^{\frac{5}{2}}}{2}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{7}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{2} lim_{x \to 4} x^{\frac{5}{2}}$

Step 2.2

Move the exponent $\frac{5}{2}$ from $x^{\frac{5}{2}}$ outside the limit using the Limits Power Rule.

$\frac{7}{2} {(lim_{x \to 4} x)}^{\frac{5}{2}}$

Step 3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{7}{2} \cdot 4^{\frac{5}{2}}$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$\frac{7}{2} \cdot {(2^{2})}^{\frac{5}{2}}$

Step 4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7}{2} \cdot 2^{2 (\frac{5}{2})}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{7}{2} \cdot 2^{2 (\frac{5}{2})}$

Step 4.3.2

Rewrite the expression.

$\frac{7}{2} \cdot 2^{5}$

Step 4.4

Raise $2$ to the power of $5$ .

$\frac{7}{2} \cdot 32$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $32$ .

$\frac{7}{2} \cdot (2 (16))$

Step 4.5.2

Cancel the common factor.

$\frac{7}{2} \cdot (2 \cdot 16)$

Step 4.5.3

Rewrite the expression.

$7 \cdot 16$

Step 4.6

Multiply $7$ by $16$ .

$112$