Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Move the limit under the radical sign.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

Simplify the answer.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.2

Add $1$ and $2$ .

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

Step 11.3

Simplify each term.

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Step 11.3.1

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

$3 \cdot (2 - 6 \cdot 64 + 8^{3})$

Step 11.3.2

Multiply $- 6$ by $64$ .

$3 \cdot (2 - 384 + 8^{3})$

Step 11.3.3

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

$3 \cdot (2 - 384 + 512)$

Step 11.4

Subtract $384$ from $2$ .

$3 \cdot (- 382 + 512)$

Step 11.5

Add $- 382$ and $512$ .

$3 \cdot 130$

Step 11.6

Multiply $3$ by $130$ .

$390$