Calculus Examples

$lim_{x \to 8} 3 x - \sqrt[3]{27 x^{3} - 2 x^{2} + x + 1}$

Step 1

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

$lim_{x \to 8} 3 x - lim_{x \to 8} \sqrt[3]{27 x^{3} - 2 x^{2} + x + 1}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - lim_{x \to 8} \sqrt[3]{27 x^{3} - 2 x^{2} + x + 1}$

Step 3

Move the limit under the radical sign.

$3 lim_{x \to 8} x - \sqrt[3]{lim_{x \to 8} 27 x^{3} - 2 x^{2} + x + 1}$

Step 4

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

$3 lim_{x \to 8} x - \sqrt[3]{lim_{x \to 8} 27 x^{3} - lim_{x \to 8} 2 x^{2} + lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 5

Move the term $27$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - \sqrt[3]{27 lim_{x \to 8} x^{3} - lim_{x \to 8} 2 x^{2} + lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 6

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

$3 lim_{x \to 8} x - \sqrt[3]{27 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 2 x^{2} + lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 7

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

$3 lim_{x \to 8} x - \sqrt[3]{27 {(lim_{x \to 8} x)}^{3} - 2 lim_{x \to 8} x^{2} + lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 8

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

$3 lim_{x \to 8} x - \sqrt[3]{27 {(lim_{x \to 8} x)}^{3} - 2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x + lim_{x \to 8} 1}$

Step 9

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

$3 lim_{x \to 8} x - \sqrt[3]{27 {(lim_{x \to 8} x)}^{3} - 2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x + 1}$

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 $x$ .

$3 \cdot 8 - \sqrt[3]{27 {(lim_{x \to 8} x)}^{3} - 2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x + 1}$

Step 10.2

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

$3 \cdot 8 - \sqrt[3]{27 \cdot 8^{3} - 2 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} x + 1}$

Step 10.3

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

$3 \cdot 8 - \sqrt[3]{27 \cdot 8^{3} - 2 \cdot 8^{2} + lim_{x \to 8} x + 1}$

Step 10.4

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

$3 \cdot 8 - \sqrt[3]{27 \cdot 8^{3} - 2 \cdot 8^{2} + 8 + 1}$

Step 11

Simplify each term.

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Step 11.1

Multiply $3$ by $8$ .

$24 - \sqrt[3]{27 \cdot 8^{3} - 2 \cdot 8^{2} + 8 + 1}$

Step 11.2

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

$24 - \sqrt[3]{27 \cdot 512 - 2 \cdot 8^{2} + 8 + 1}$

Step 11.3

Multiply $27$ by $512$ .

$24 - \sqrt[3]{13824 - 2 \cdot 8^{2} + 8 + 1}$

Step 11.4

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

$24 - \sqrt[3]{13824 - 2 \cdot 64 + 8 + 1}$

Step 11.5

Multiply $- 2$ by $64$ .

$24 - \sqrt[3]{13824 - 128 + 8 + 1}$

Step 11.6

Subtract $128$ from $13824$ .

$24 - \sqrt[3]{13696 + 8 + 1}$

Step 11.7

Add $13696$ and $8$ .

$24 - \sqrt[3]{13704 + 1}$

Step 11.8

Add $13704$ and $1$ .

$24 - \sqrt[3]{13705}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$24 - \sqrt[3]{13705}$

Decimal Form:

$0.06906429 \dots$