Calculus Examples

$lim_{x \to 8} \frac{\sqrt[3]{x^{3} - 4 x^{2} - 7 x - 5}}{3} \cdot (16 x - 8)$

Step 1

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

$lim_{x \to 8} \frac{\sqrt[3]{x^{3} - 4 x^{2} - 7 x - 5}}{3} \cdot lim_{x \to 8} 16 x - 8$

Step 2

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 8} \sqrt[3]{x^{3} - 4 x^{2} - 7 x - 5} \cdot lim_{x \to 8} 16 x - 8$

Step 3

Move the limit under the radical sign.

$\frac{1}{3} \sqrt[3]{lim_{x \to 8} x^{3} - 4 x^{2} - 7 x - 5} \cdot lim_{x \to 8} 16 x - 8$

Step 4

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

$\frac{1}{3} \sqrt[3]{lim_{x \to 8} x^{3} - lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 7 x - lim_{x \to 8} 5} \cdot lim_{x \to 8} 16 x - 8$

Step 5

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

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 4 x^{2} - lim_{x \to 8} 7 x - lim_{x \to 8} 5} \cdot lim_{x \to 8} 16 x - 8$

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 lim_{x \to 8} x^{2} - lim_{x \to 8} 7 x - lim_{x \to 8} 5} \cdot lim_{x \to 8} 16 x - 8$

Step 7

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

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 7 x - lim_{x \to 8} 5} \cdot lim_{x \to 8} 16 x - 8$

Step 8

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - lim_{x \to 8} 5} \cdot lim_{x \to 8} 16 x - 8$

Step 9

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot lim_{x \to 8} 16 x - 8$

Step 10

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

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot (lim_{x \to 8} 16 x - lim_{x \to 8} 8)$

Step 11

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

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot (16 lim_{x \to 8} x - lim_{x \to 8} 8)$

Step 12

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

$\frac{1}{3} \sqrt[3]{{(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot (16 lim_{x \to 8} x - 1 \cdot 8)$

Step 13

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

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Step 13.1

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

$\frac{1}{3} \sqrt[3]{8^{3} - 4 {(lim_{x \to 8} x)}^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot (16 lim_{x \to 8} x - 1 \cdot 8)$

Step 13.2

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

$\frac{1}{3} \sqrt[3]{8^{3} - 4 \cdot 8^{2} - 7 lim_{x \to 8} x - 1 \cdot 5} \cdot (16 lim_{x \to 8} x - 1 \cdot 8)$

Step 13.3

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

$\frac{1}{3} \sqrt[3]{8^{3} - 4 \cdot 8^{2} - 7 \cdot 8 - 1 \cdot 5} \cdot (16 lim_{x \to 8} x - 1 \cdot 8)$

Step 13.4

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

$\frac{1}{3} \sqrt[3]{8^{3} - 4 \cdot 8^{2} - 7 \cdot 8 - 1 \cdot 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14

Simplify the answer.

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Step 14.1

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

$\frac{1}{3} \sqrt[3]{512 - 4 \cdot 8^{2} - 7 \cdot 8 - 1 \cdot 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.2

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

$\frac{1}{3} \sqrt[3]{512 - 4 \cdot 64 - 7 \cdot 8 - 1 \cdot 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.3

Multiply $- 4$ by $64$ .

$\frac{1}{3} \sqrt[3]{512 - 256 - 7 \cdot 8 - 1 \cdot 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.4

Multiply $- 7$ by $8$ .

$\frac{1}{3} \sqrt[3]{512 - 256 - 56 - 1 \cdot 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.5

Multiply $- 1$ by $5$ .

$\frac{1}{3} \sqrt[3]{512 - 256 - 56 - 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.6

Subtract $256$ from $512$ .

$\frac{1}{3} \sqrt[3]{256 - 56 - 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.7

Subtract $56$ from $256$ .

$\frac{1}{3} \sqrt[3]{200 - 5} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.8

Subtract $5$ from $200$ .

$\frac{1}{3} \sqrt[3]{195} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.9

Combine $\frac{1}{3}$ and $\sqrt[3]{195}$ .

$\frac{\sqrt[3]{195}}{3} \cdot (16 \cdot 8 - 1 \cdot 8)$

Step 14.10

Simplify each term.

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Step 14.10.1

Multiply $16$ by $8$ .

$\frac{\sqrt[3]{195}}{3} \cdot (128 - 1 \cdot 8)$

Step 14.10.2

Multiply $- 1$ by $8$ .

$\frac{\sqrt[3]{195}}{3} \cdot (128 - 8)$

Step 14.11

Subtract $8$ from $128$ .

$\frac{\sqrt[3]{195}}{3} \cdot 120$

Step 14.12

Cancel the common factor of $3$ .

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Step 14.12.1

Factor $3$ out of $120$ .

$\frac{\sqrt[3]{195}}{3} \cdot (3 (40))$

Step 14.12.2

Cancel the common factor.

$\frac{\sqrt[3]{195}}{3} \cdot (3 \cdot 40)$

Step 14.12.3

Rewrite the expression.

$\sqrt[3]{195} \cdot 40$

Step 14.13

Move $40$ to the left of $\sqrt[3]{195}$ .

$40 \sqrt[3]{195}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$40 \sqrt[3]{195}$

Decimal Form:

$231.95559990 \dots$