Calculus Examples

$\frac{lim_{n \to 8} {(2 n)}^{3} + 12 n^{2} - 20}{12 n^{2} + n}$

Step 1

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

$\frac{lim_{n \to 8} {(2 n)}^{3} + lim_{n \to 8} 12 n^{2} - lim_{n \to 8} 20}{12 n^{2} + n}$

Step 2

Move the exponent $3$ from ${(2 n)}^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{n \to 8} 2 n)}^{3} + lim_{n \to 8} 12 n^{2} - lim_{n \to 8} 20}{12 n^{2} + n}$

Step 3

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

$\frac{{(2 lim_{n \to 8} n)}^{3} + lim_{n \to 8} 12 n^{2} - lim_{n \to 8} 20}{12 n^{2} + n}$

Step 4

Move the term $12$ outside of the limit because it is constant with respect to $n$ .

$\frac{{(2 lim_{n \to 8} n)}^{3} + 12 lim_{n \to 8} n^{2} - lim_{n \to 8} 20}{12 n^{2} + n}$

Step 5

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

$\frac{{(2 lim_{n \to 8} n)}^{3} + 12 {(lim_{n \to 8} n)}^{2} - lim_{n \to 8} 20}{12 n^{2} + n}$

Step 6

Evaluate the limit of $20$ which is constant as $n$ approaches $8$ .

$\frac{{(2 lim_{n \to 8} n)}^{3} + 12 {(lim_{n \to 8} n)}^{2} - 1 \cdot 20}{12 n^{2} + n}$

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 $n$ .

$\frac{{(2 \cdot 8)}^{3} + 12 {(lim_{n \to 8} n)}^{2} - 1 \cdot 20}{12 n^{2} + n}$

Step 7.2

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

$\frac{{(2 \cdot 8)}^{3} + 12 \cdot 8^{2} - 1 \cdot 20}{12 n^{2} + n}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $2$ by $8$ .

$\frac{16^{3} + 12 \cdot 8^{2} - 1 \cdot 20}{12 n^{2} + n}$

Step 8.1.2

Raise $16$ to the power of $3$ .

$\frac{4096 + 12 \cdot 8^{2} - 1 \cdot 20}{12 n^{2} + n}$

Step 8.1.3

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

$\frac{4096 + 12 \cdot 64 - 1 \cdot 20}{12 n^{2} + n}$

Step 8.1.4

Multiply $12$ by $64$ .

$\frac{4096 + 768 - 1 \cdot 20}{12 n^{2} + n}$

Step 8.1.5

Multiply $- 1$ by $20$ .

$\frac{4096 + 768 - 20}{12 n^{2} + n}$

Step 8.1.6

Add $4096$ and $768$ .

$\frac{4864 - 20}{12 n^{2} + n}$

Step 8.1.7

Subtract $20$ from $4864$ .

$\frac{4844}{12 n^{2} + n}$

Step 8.2

Factor $n$ out of $12 n^{2} + n$ .

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Step 8.2.1

Factor $n$ out of $12 n^{2}$ .

$\frac{4844}{n (12 n) + n}$

Step 8.2.2

Raise $n$ to the power of $1$ .

$\frac{4844}{n (12 n) + n^{1}}$

Step 8.2.3

Factor $n$ out of $n^{1}$ .

$\frac{4844}{n (12 n) + n \cdot 1}$

Step 8.2.4

Factor $n$ out of $n (12 n) + n \cdot 1$ .

$\frac{4844}{n (12 n + 1)}$