Calculus Examples

$lim_{x \to 8} (\frac{4}{3 x^{3}}) (2 x \cdot x \cdot x + 3 x \cdot x + x)$

Step 1

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

$lim_{x \to 8} \frac{4}{3 x^{3}} \cdot lim_{x \to 8} 2 x \cdot x \cdot x + 3 x \cdot x + x$

Step 2

Move the term $\frac{4}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{4}{3} lim_{x \to 8} \frac{1}{x^{3}} \cdot lim_{x \to 8} 2 x \cdot x \cdot x + 3 x \cdot x + x$

Step 3

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

$\frac{4}{3} \cdot \frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{3}} \cdot lim_{x \to 8} 2 x \cdot x \cdot x + 3 x \cdot x + x$

Step 4

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

$\frac{4}{3} \cdot \frac{1}{lim_{x \to 8} x^{3}} \cdot lim_{x \to 8} 2 x \cdot x \cdot x + 3 x \cdot x + x$

Step 5

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot lim_{x \to 8} 2 x \cdot x \cdot x + 3 x \cdot x + x$

Step 6

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot (lim_{x \to 8} 2 x \cdot x \cdot x + lim_{x \to 8} 3 x \cdot x + lim_{x \to 8} x)$

Step 7

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot (2 lim_{x \to 8} x \cdot x \cdot x + lim_{x \to 8} 3 x \cdot x + lim_{x \to 8} x)$

Step 8

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot (2 lim_{x \to 8} x \cdot lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} 3 x \cdot x + lim_{x \to 8} x)$

Step 9

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot (2 lim_{x \to 8} x \cdot lim_{x \to 8} x \cdot lim_{x \to 8} x + 3 lim_{x \to 8} x \cdot x + lim_{x \to 8} x)$

Step 10

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

$\frac{4}{3} \cdot \frac{1}{{(lim_{x \to 8} x)}^{3}} \cdot (2 lim_{x \to 8} x \cdot lim_{x \to 8} x \cdot lim_{x \to 8} x + 3 lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11

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

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Step 11.1

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 lim_{x \to 8} x \cdot lim_{x \to 8} x \cdot lim_{x \to 8} x + 3 lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11.2

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot lim_{x \to 8} x \cdot lim_{x \to 8} x + 3 lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11.3

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot lim_{x \to 8} x + 3 lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11.4

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 lim_{x \to 8} x \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11.5

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot lim_{x \to 8} x + lim_{x \to 8} x)$

Step 11.6

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + lim_{x \to 8} x)$

Step 11.7

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

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

$\frac{4}{3} \cdot \frac{1}{8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12

Simplify the answer.

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Step 12.1

Combine.

$\frac{4 \cdot 1}{3 \cdot 8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.2

Multiply $4$ by $1$ .

$\frac{4}{3 \cdot 8^{3}} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.3

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

$\frac{4}{3 \cdot 512} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.4

Multiply $3$ by $512$ .

$\frac{4}{1536} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.5

Cancel the common factor of $4$ and $1536$ .

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Step 12.5.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{1536} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.5.2

Cancel the common factors.

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Step 12.5.2.1

Factor $4$ out of $1536$ .

$\frac{4 \cdot 1}{4 \cdot 384} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.5.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 384} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.5.2.3

Rewrite the expression.

$\frac{1}{384} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

$\frac{1}{384} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

$\frac{1}{384} \cdot (2 \cdot 8 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.6

Simplify each term.

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Step 12.6.1

Multiply $2 \cdot 8 \cdot 8 \cdot 8$ .

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Step 12.6.1.1

Multiply $2$ by $8$ .

$\frac{1}{384} \cdot (16 \cdot 8 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.6.1.2

Multiply $16$ by $8$ .

$\frac{1}{384} \cdot (128 \cdot 8 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.6.1.3

Multiply $128$ by $8$ .

$\frac{1}{384} \cdot (1024 + 3 \cdot 8 \cdot 8 + 8)$

$\frac{1}{384} \cdot (1024 + 3 \cdot 8 \cdot 8 + 8)$

Step 12.6.2

Multiply $3 \cdot 8 \cdot 8$ .

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Step 12.6.2.1

Multiply $3$ by $8$ .

$\frac{1}{384} \cdot (1024 + 24 \cdot 8 + 8)$

Step 12.6.2.2

Multiply $24$ by $8$ .

$\frac{1}{384} \cdot (1024 + 192 + 8)$

$\frac{1}{384} \cdot (1024 + 192 + 8)$

$\frac{1}{384} \cdot (1024 + 192 + 8)$

Step 12.7

Add $1024$ and $192$ .

$\frac{1}{384} \cdot (1216 + 8)$

Step 12.8

Add $1216$ and $8$ .

$\frac{1}{384} \cdot 1224$

Step 12.9

Cancel the common factor of $24$ .

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Step 12.9.1

Factor $24$ out of $384$ .

$\frac{1}{24 (16)} \cdot 1224$

Step 12.9.2

Factor $24$ out of $1224$ .

$\frac{1}{24 \cdot 16} \cdot (24 \cdot 51)$

Step 12.9.3

Cancel the common factor.

$\frac{1}{24 \cdot 16} \cdot (24 \cdot 51)$

Step 12.9.4

Rewrite the expression.

$\frac{1}{16} \cdot 51$

$\frac{1}{16} \cdot 51$

Step 12.10

Combine $\frac{1}{16}$ and $51$ .

$\frac{51}{16}$

$\frac{51}{16}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{51}{16}$

Decimal Form:

$3.1875$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$