Calculus Examples

$lim_{x \to 8} \frac{4 x^{3} + 8}{4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 1

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

$\frac{lim_{x \to 8} 4 x^{3} + 8}{lim_{x \to 8} 4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 2

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

$\frac{lim_{x \to 8} 4 x^{3} + lim_{x \to 8} 8}{lim_{x \to 8} 4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 3

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

$\frac{4 lim_{x \to 8} x^{3} + lim_{x \to 8} 8}{lim_{x \to 8} 4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 4

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 8}{lim_{x \to 8} 4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 5

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{lim_{x \to 8} 4 x^{3} + \sqrt{64 x^{6} + 1}}$

Step 6

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{lim_{x \to 8} 4 x^{3} + lim_{x \to 8} \sqrt{64 x^{6} + 1}}$

Step 7

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 lim_{x \to 8} x^{3} + lim_{x \to 8} \sqrt{64 x^{6} + 1}}$

Step 8

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} \sqrt{64 x^{6} + 1}}$

Step 9

Move the limit under the radical sign.

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{lim_{x \to 8} 64 x^{6} + 1}}$

Step 10

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{lim_{x \to 8} 64 x^{6} + lim_{x \to 8} 1}}$

Step 11

Move the term $64$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{64 lim_{x \to 8} x^{6} + lim_{x \to 8} 1}}$

Step 12

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{64 {(lim_{x \to 8} x)}^{6} + lim_{x \to 8} 1}}$

Step 13

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

$\frac{4 {(lim_{x \to 8} x)}^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{64 {(lim_{x \to 8} x)}^{6} + 1}}$

Step 14

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

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Step 14.1

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

$\frac{4 \cdot 8^{3} + 8}{4 {(lim_{x \to 8} x)}^{3} + \sqrt{64 {(lim_{x \to 8} x)}^{6} + 1}}$

Step 14.2

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

$\frac{4 \cdot 8^{3} + 8}{4 \cdot 8^{3} + \sqrt{64 {(lim_{x \to 8} x)}^{6} + 1}}$

Step 14.3

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

$\frac{4 \cdot 8^{3} + 8}{4 \cdot 8^{3} + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15

Simplify the answer.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

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

$\frac{4 \cdot 512 + 8}{4 \cdot 8^{3} + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15.1.2

Multiply $4$ by $512$ .

$\frac{2048 + 8}{4 \cdot 8^{3} + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15.1.3

Add $2048$ and $8$ .

$\frac{2056}{4 \cdot 8^{3} + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15.2

Simplify the denominator.

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Step 15.2.1

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

$\frac{2056}{4 \cdot 512 + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15.2.2

Multiply $4$ by $512$ .

$\frac{2056}{2048 + \sqrt{64 \cdot 8^{6} + 1}}$

Step 15.2.3

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

$\frac{2056}{2048 + \sqrt{64 \cdot 262144 + 1}}$

Step 15.2.4

Multiply $64$ by $262144$ .

$\frac{2056}{2048 + \sqrt{16777216 + 1}}$

Step 15.2.5

Add $16777216$ and $1$ .

$\frac{2056}{2048 + \sqrt{16777217}}$

Step 15.3

Multiply $\frac{2056}{2048 + \sqrt{16777217}}$ by $\frac{2048 - \sqrt{16777217}}{2048 - \sqrt{16777217}}$ .

$\frac{2056}{2048 + \sqrt{16777217}} \cdot \frac{2048 - \sqrt{16777217}}{2048 - \sqrt{16777217}}$

Step 15.4

Multiply $\frac{2056}{2048 + \sqrt{16777217}}$ by $\frac{2048 - \sqrt{16777217}}{2048 - \sqrt{16777217}}$ .

$\frac{2056 (2048 - \sqrt{16777217})}{(2048 + \sqrt{16777217}) (2048 - \sqrt{16777217})}$

Step 15.5

Expand the denominator using the FOIL method.

$\frac{2056 (2048 - \sqrt{16777217})}{4194304 - 2048 \sqrt{16777217} + \sqrt{16777217} \cdot 2048 - {\sqrt{16777217}}^{2}}$

Step 15.6

Simplify.

$\frac{2056 (2048 - \sqrt{16777217})}{- 12582913}$

Step 15.7

Move the negative in front of the fraction.

$- \frac{2056 (2048 - \sqrt{16777217})}{12582913}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$- \frac{2056 (2048 - \sqrt{16777217})}{12582913}$

Decimal Form:

$0.33463541 \dots$