Calculus Examples

$lim_{n \to 8} \frac{n}{10} + \frac{10}{n}$

Step 1

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

$lim_{n \to 8} \frac{n}{10} + lim_{n \to 8} \frac{10}{n}$

Step 2

Move the term $\frac{1}{10}$ outside of the limit because it is constant with respect to $n$ .

$\frac{1}{10} lim_{n \to 8} n + lim_{n \to 8} \frac{10}{n}$

Step 3

Move the term $10$ outside of the limit because it is constant with respect to $n$ .

$\frac{1}{10} lim_{n \to 8} n + 10 lim_{n \to 8} \frac{1}{n}$

Step 4

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

$\frac{1}{10} lim_{n \to 8} n + 10 \frac{lim_{n \to 8} 1}{lim_{n \to 8} n}$

Step 5

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

$\frac{1}{10} lim_{n \to 8} n + 10 \frac{1}{lim_{n \to 8} n}$

Step 6

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

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Step 6.1

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

$\frac{1}{10} \cdot 8 + 10 \frac{1}{lim_{n \to 8} n}$

Step 6.2

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

$\frac{1}{10} \cdot 8 + 10 (\frac{1}{8})$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Cancel the common factor of $2$ .

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Step 7.1.1.1

Factor $2$ out of $10$ .

$\frac{1}{2 (5)} \cdot 8 + 10 (\frac{1}{8})$

Step 7.1.1.2

Factor $2$ out of $8$ .

$\frac{1}{2 \cdot 5} \cdot (2 \cdot 4) + 10 (\frac{1}{8})$

Step 7.1.1.3

Cancel the common factor.

$\frac{1}{2 \cdot 5} \cdot (2 \cdot 4) + 10 (\frac{1}{8})$

Step 7.1.1.4

Rewrite the expression.

$\frac{1}{5} \cdot 4 + 10 (\frac{1}{8})$

Step 7.1.2

Combine $\frac{1}{5}$ and $4$ .

$\frac{4}{5} + 10 (\frac{1}{8})$

Step 7.1.3

Cancel the common factor of $2$ .

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Step 7.1.3.1

Factor $2$ out of $10$ .

$\frac{4}{5} + 2 (5) \frac{1}{8}$

Step 7.1.3.2

Factor $2$ out of $8$ .

$\frac{4}{5} + 2 \cdot 5 \frac{1}{2 \cdot 4}$

Step 7.1.3.3

Cancel the common factor.

$\frac{4}{5} + 2 \cdot 5 \frac{1}{2 \cdot 4}$

Step 7.1.3.4

Rewrite the expression.

$\frac{4}{5} + 5 (\frac{1}{4})$

Step 7.1.4

Combine $5$ and $\frac{1}{4}$ .

$\frac{4}{5} + \frac{5}{4}$

Step 7.2

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4}{5} \cdot \frac{4}{4} + \frac{5}{4}$

Step 7.3

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{4}{5} \cdot \frac{4}{4} + \frac{5}{4} \cdot \frac{5}{5}$

Step 7.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.4.1

Multiply $\frac{4}{5}$ by $\frac{4}{4}$ .

$\frac{4 \cdot 4}{5 \cdot 4} + \frac{5}{4} \cdot \frac{5}{5}$

Step 7.4.2

Multiply $5$ by $4$ .

$\frac{4 \cdot 4}{20} + \frac{5}{4} \cdot \frac{5}{5}$

Step 7.4.3

Multiply $\frac{5}{4}$ by $\frac{5}{5}$ .

$\frac{4 \cdot 4}{20} + \frac{5 \cdot 5}{4 \cdot 5}$

Step 7.4.4

Multiply $4$ by $5$ .

$\frac{4 \cdot 4}{20} + \frac{5 \cdot 5}{20}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{4 \cdot 4 + 5 \cdot 5}{20}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $4$ by $4$ .

$\frac{16 + 5 \cdot 5}{20}$

Step 7.6.2

Multiply $5$ by $5$ .

$\frac{16 + 25}{20}$

Step 7.6.3

Add $16$ and $25$ .

$\frac{41}{20}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{41}{20}$

Decimal Form:

$2.05$