Calculus Examples

$lim_{n \to 8} \frac{(n + 1) (2 n + 1)}{6 n^{2}}$

Step 1

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $n$ .

$\frac{1}{6} lim_{n \to 8} \frac{(n + 1) (2 n + 1)}{n^{2}}$

Step 2

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

$\frac{1}{6} \cdot \frac{lim_{n \to 8} (n + 1) (2 n + 1)}{lim_{n \to 8} n^{2}}$

Step 3

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

$\frac{1}{6} \cdot \frac{lim_{n \to 8} n + 1 \cdot lim_{n \to 8} 2 n + 1}{lim_{n \to 8} n^{2}}$

Step 4

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + lim_{n \to 8} 1) \cdot lim_{n \to 8} 2 n + 1}{lim_{n \to 8} n^{2}}$

Step 5

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + 1) \cdot lim_{n \to 8} 2 n + 1}{lim_{n \to 8} n^{2}}$

Step 6

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + 1) \cdot (lim_{n \to 8} 2 n + lim_{n \to 8} 1)}{lim_{n \to 8} n^{2}}$

Step 7

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + 1) \cdot (2 lim_{n \to 8} n + lim_{n \to 8} 1)}{lim_{n \to 8} n^{2}}$

Step 8

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + 1) \cdot (2 lim_{n \to 8} n + 1)}{lim_{n \to 8} n^{2}}$

Step 9

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

$\frac{1}{6} \cdot \frac{(lim_{n \to 8} n + 1) \cdot (2 lim_{n \to 8} n + 1)}{{(lim_{n \to 8} n)}^{2}}$

Step 10

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

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Step 10.1

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

$\frac{1}{6} \cdot \frac{(8 + 1) \cdot (2 lim_{n \to 8} n + 1)}{{(lim_{n \to 8} n)}^{2}}$

Step 10.2

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

$\frac{1}{6} \cdot \frac{(8 + 1) \cdot (2 \cdot 8 + 1)}{{(lim_{n \to 8} n)}^{2}}$

Step 10.3

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

$\frac{1}{6} \cdot \frac{(8 + 1) \cdot (2 \cdot 8 + 1)}{8^{2}}$

Step 11

Simplify the answer.

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Step 11.1

Combine.

$\frac{1 ((8 + 1) \cdot (2 \cdot 8 + 1))}{6 \cdot 8^{2}}$

Step 11.2

Multiply $8 + 1$ by $1$ .

$\frac{(8 + 1) (2 \cdot 8 + 1)}{6 \cdot 8^{2}}$

Step 11.3

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

$\frac{(8 + 1) (2 \cdot 8 + 1)}{6 \cdot 64}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Add $8$ and $1$ .

$\frac{9 (2 \cdot 8 + 1)}{6 \cdot 64}$

Step 11.4.2

Multiply $2$ by $8$ .

$\frac{9 (16 + 1)}{6 \cdot 64}$

Step 11.4.3

Add $16$ and $1$ .

$\frac{9 \cdot 17}{6 \cdot 64}$

Step 11.5

Multiply $6$ by $64$ .

$\frac{9 \cdot 17}{384}$

Step 11.6

Multiply $9$ by $17$ .

$\frac{153}{384}$

Step 11.7

Cancel the common factor of $153$ and $384$ .

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Step 11.7.1

Factor $3$ out of $153$ .

$\frac{3 (51)}{384}$

Step 11.7.2

Cancel the common factors.

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Step 11.7.2.1

Factor $3$ out of $384$ .

$\frac{3 \cdot 51}{3 \cdot 128}$

Step 11.7.2.2

Cancel the common factor.

$\frac{3 \cdot 51}{3 \cdot 128}$

Step 11.7.2.3

Rewrite the expression.

$\frac{51}{128}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{51}{128}$

Decimal Form:

$0.3984375$