Calculus Examples

$lim_{x \to 8} (1 + \sqrt[3]{x}) (2 - 6 x^{x^{3}})$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to 8} 1 + \sqrt[3]{x} \cdot lim_{x \to 8} 2 - 6 x^{x^{3}}$

Step 1.2

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

$(lim_{x \to 8} 1 + lim_{x \to 8} \sqrt[3]{x}) \cdot lim_{x \to 8} 2 - 6 x^{x^{3}}$

Step 1.3

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

$(1 + lim_{x \to 8} \sqrt[3]{x}) \cdot lim_{x \to 8} 2 - 6 x^{x^{3}}$

Step 1.4

Move the limit under the radical sign.

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot lim_{x \to 8} 2 - 6 x^{x^{3}}$

Step 1.5

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

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (lim_{x \to 8} 2 - lim_{x \to 8} 6 x^{x^{3}})$

Step 1.6

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - lim_{x \to 8} 6 x^{x^{3}})$

Step 1.7

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 lim_{x \to 8} x^{x^{3}})$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{x^{3}}$ as $e^{\ln (x^{x^{3}})}$ .

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 lim_{x \to 8} e^{\ln (x^{x^{3}})})$

Step 2.2

Expand $\ln (x^{x^{3}})$ by moving $x^{3}$ outside the logarithm.

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 lim_{x \to 8} e^{x^{3} \ln (x)})$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 e^{lim_{x \to 8} x^{3} \ln (x)})$

Step 3.2

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

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 e^{lim_{x \to 8} x^{3} \cdot lim_{x \to 8} \ln (x)})$

Step 3.3

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

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 e^{{(lim_{x \to 8} x)}^{3} \cdot lim_{x \to 8} \ln (x)})$

Step 3.4

Move the limit inside the logarithm.

$(1 + \sqrt[3]{lim_{x \to 8} x}) \cdot (2 - 6 e^{{(lim_{x \to 8} x)}^{3} \cdot \ln (lim_{x \to 8} x)})$

Step 4

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

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Step 4.1

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

$(1 + \sqrt[3]{8}) \cdot (2 - 6 e^{{(lim_{x \to 8} x)}^{3} \cdot \ln (lim_{x \to 8} x)})$

Step 4.2

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

$(1 + \sqrt[3]{8}) \cdot (2 - 6 e^{8^{3} \cdot \ln (lim_{x \to 8} x)})$

Step 4.3

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

$(1 + \sqrt[3]{8}) \cdot (2 - 6 e^{8^{3} \cdot \ln (8)})$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite $8$ as $2^{3}$ .

$(1 + \sqrt[3]{2^{3}}) \cdot (2 - 6 e^{8^{3} \cdot \ln (8)})$

Step 5.1.2

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

$(1 + 2) \cdot (2 - 6 e^{8^{3} \cdot \ln (8)})$

Step 5.2

Add $1$ and $2$ .

$3 \cdot (2 - 6 e^{8^{3} \cdot \ln (8)})$

Step 5.3

Simplify each term.

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Step 5.3.1

Raise $8$ to the power of $3$ .

$3 \cdot (2 - 6 e^{512 \cdot \ln (8)})$

Step 5.3.2

Simplify $512 \cdot \ln (8)$ by moving $512$ inside the logarithm.

$3 \cdot (2 - 6 e^{\ln (8^{512})})$

Step 5.3.3

Exponentiation and log are inverse functions.

$3 \cdot (2 - 6 \cdot 8^{512})$

Step 5.4

Apply the distributive property.

$3 \cdot 2 + 3 (- 6 \cdot 8^{512})$

Step 5.5

Multiply $3$ by $2$ .

$6 + 3 (- 6 \cdot 8^{512})$

Step 5.6

Multiply $- 6$ by $3$ .

$6 - 18 \cdot 8^{512}$