Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Simplify the answer.

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Step 8.1

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

$2 \frac{512}{8^{2} + 4 \cdot 8}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

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

$2 \frac{512}{64 + 4 \cdot 8}$

Step 8.2.2

Multiply $4$ by $8$ .

$2 \frac{512}{64 + 32}$

Step 8.2.3

Add $64$ and $32$ .

$2 (\frac{512}{96})$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Factor $2$ out of $96$ .

$2 \frac{512}{2 (48)}$

Step 8.3.2

Cancel the common factor.

$2 \frac{512}{2 \cdot 48}$

Step 8.3.3

Rewrite the expression.

$\frac{512}{48}$

Step 8.4

Cancel the common factor of $512$ and $48$ .

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Step 8.4.1

Factor $16$ out of $512$ .

$\frac{16 (32)}{48}$

Step 8.4.2

Cancel the common factors.

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Step 8.4.2.1

Factor $16$ out of $48$ .

$\frac{16 \cdot 32}{16 \cdot 3}$

Step 8.4.2.2

Cancel the common factor.

$\frac{16 \cdot 32}{16 \cdot 3}$

Step 8.4.2.3

Rewrite the expression.

$\frac{32}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{3}$

Decimal Form:

$10.\overline{6}$