Calculus Examples

$lim_{x \to 8} x^{3} - \sqrt{x^{6} - 2 x^{3} - 1}$

Step 1

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

$lim_{x \to 8} x^{3} - lim_{x \to 8} \sqrt{x^{6} - 2 x^{3} - 1}$

Step 2

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

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

Move the exponent $6$ from $x^{6}$ outside the limit using the Limits Power Rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

$8^{3} - \sqrt{8^{6} - 2 \cdot 8^{3} - 1 \cdot 1}$

Step 10

Simplify each term.

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Step 10.1

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

$512 - \sqrt{8^{6} - 2 \cdot 8^{3} - 1 \cdot 1}$

Step 10.2

Raise $8$ to the power of $6$ .

$512 - \sqrt{262144 - 2 \cdot 8^{3} - 1 \cdot 1}$

Step 10.3

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

$512 - \sqrt{262144 - 2 \cdot 512 - 1 \cdot 1}$

Step 10.4

Multiply $- 2$ by $512$ .

$512 - \sqrt{262144 - 1024 - 1 \cdot 1}$

Step 10.5

Multiply $- 1$ by $1$ .

$512 - \sqrt{262144 - 1024 - 1}$

Step 10.6

Subtract $1024$ from $262144$ .

$512 - \sqrt{261120 - 1}$

Step 10.7

Subtract $1$ from $261120$ .

$512 - \sqrt{261119}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$512 - \sqrt{261119}$

Decimal Form:

$1.00195695 \dots$