Calculus Examples

$lim_{n \to 8} \frac{1 + 2 + 3 + n}{3 n + 1} - \frac{n}{6}$

Step 1

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

$lim_{n \to 8} \frac{1 + 2 + 3 + n}{3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 2

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

$\frac{lim_{n \to 8} 1 + 2 + 3 + n}{lim_{n \to 8} 3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 3

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

$\frac{lim_{n \to 8} 1 + lim_{n \to 8} 2 + lim_{n \to 8} 3 + lim_{n \to 8} n}{lim_{n \to 8} 3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 4

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

$\frac{1 + lim_{n \to 8} 2 + lim_{n \to 8} 3 + lim_{n \to 8} n}{lim_{n \to 8} 3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 5

Evaluate the limit of $2$ which is constant as $n$ approaches $8$ .

$\frac{1 + 2 + lim_{n \to 8} 3 + lim_{n \to 8} n}{lim_{n \to 8} 3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 6

Evaluate the limit of $3$ which is constant as $n$ approaches $8$ .

$\frac{1 + 2 + 3 + lim_{n \to 8} n}{lim_{n \to 8} 3 n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 7

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

$\frac{1 + 2 + 3 + lim_{n \to 8} n}{lim_{n \to 8} 3 n + lim_{n \to 8} 1} - lim_{n \to 8} \frac{n}{6}$

Step 8

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

$\frac{1 + 2 + 3 + lim_{n \to 8} n}{3 lim_{n \to 8} n + lim_{n \to 8} 1} - lim_{n \to 8} \frac{n}{6}$

Step 9

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

$\frac{1 + 2 + 3 + lim_{n \to 8} n}{3 lim_{n \to 8} n + 1} - lim_{n \to 8} \frac{n}{6}$

Step 10

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

$\frac{1 + 2 + 3 + lim_{n \to 8} n}{3 lim_{n \to 8} n + 1} - \frac{1}{6} lim_{n \to 8} n$

Step 11

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

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Step 11.1

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

$\frac{1 + 2 + 3 + 8}{3 lim_{n \to 8} n + 1} - \frac{1}{6} lim_{n \to 8} n$

Step 11.2

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

$\frac{1 + 2 + 3 + 8}{3 \cdot 8 + 1} - \frac{1}{6} lim_{n \to 8} n$

Step 11.3

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

$\frac{1 + 2 + 3 + 8}{3 \cdot 8 + 1} - \frac{1}{6} \cdot 8$

Step 12

Simplify the answer.

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Step 12.1

Add $1$ and $2$ .

$\frac{3 + 3 + 8}{3 \cdot 8 + 1} - \frac{1}{6} \cdot 8$

Step 12.2

Add $3$ and $3$ .

$\frac{6 + 8}{3 \cdot 8 + 1} - \frac{1}{6} \cdot 8$

Step 12.3

Add $6$ and $8$ .

$\frac{14}{3 \cdot 8 + 1} - \frac{1}{6} \cdot 8$

Step 12.4

Simplify each term.

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Step 12.4.1

Simplify the denominator.

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Step 12.4.1.1

Multiply $3$ by $8$ .

$\frac{14}{24 + 1} - \frac{1}{6} \cdot 8$

Step 12.4.1.2

Add $24$ and $1$ .

$\frac{14}{25} - \frac{1}{6} \cdot 8$

Step 12.4.2

Cancel the common factor of $2$ .

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Step 12.4.2.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$\frac{14}{25} + \frac{- 1}{6} \cdot 8$

Step 12.4.2.2

Factor $2$ out of $6$ .

$\frac{14}{25} + \frac{- 1}{2 (3)} \cdot 8$

Step 12.4.2.3

Factor $2$ out of $8$ .

$\frac{14}{25} + \frac{- 1}{2 \cdot 3} \cdot (2 \cdot 4)$

Step 12.4.2.4

Cancel the common factor.

$\frac{14}{25} + \frac{- 1}{2 \cdot 3} \cdot (2 \cdot 4)$

Step 12.4.2.5

Rewrite the expression.

$\frac{14}{25} + \frac{- 1}{3} \cdot 4$

Step 12.4.3

Combine $\frac{- 1}{3}$ and $4$ .

$\frac{14}{25} + \frac{- 1 \cdot 4}{3}$

Step 12.4.4

Multiply $- 1$ by $4$ .

$\frac{14}{25} + \frac{- 4}{3}$

Step 12.4.5

Move the negative in front of the fraction.

$\frac{14}{25} - \frac{4}{3}$

Step 12.5

To write $\frac{14}{25}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{14}{25} \cdot \frac{3}{3} - \frac{4}{3}$

Step 12.6

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{14}{25} \cdot \frac{3}{3} - \frac{4}{3} \cdot \frac{25}{25}$

Step 12.7

Write each expression with a common denominator of $75$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.7.1

Multiply $\frac{14}{25}$ by $\frac{3}{3}$ .

$\frac{14 \cdot 3}{25 \cdot 3} - \frac{4}{3} \cdot \frac{25}{25}$

Step 12.7.2

Multiply $25$ by $3$ .

$\frac{14 \cdot 3}{75} - \frac{4}{3} \cdot \frac{25}{25}$

Step 12.7.3

Multiply $\frac{4}{3}$ by $\frac{25}{25}$ .

$\frac{14 \cdot 3}{75} - \frac{4 \cdot 25}{3 \cdot 25}$

Step 12.7.4

Multiply $3$ by $25$ .

$\frac{14 \cdot 3}{75} - \frac{4 \cdot 25}{75}$

Step 12.8

Combine the numerators over the common denominator.

$\frac{14 \cdot 3 - 4 \cdot 25}{75}$

Step 12.9

Simplify the numerator.

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Step 12.9.1

Multiply $14$ by $3$ .

$\frac{42 - 4 \cdot 25}{75}$

Step 12.9.2

Multiply $- 4$ by $25$ .

$\frac{42 - 100}{75}$

Step 12.9.3

Subtract $100$ from $42$ .

$\frac{- 58}{75}$

Step 12.10

Move the negative in front of the fraction.

$- \frac{58}{75}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{58}{75}$

Decimal Form:

$- 0.77\overline{3}$