Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

$\frac{1}{8} - \frac{1}{8 + 1}$

Step 9

Simplify the answer.

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Step 9.1

Add $8$ and $1$ .

$\frac{1}{8} - \frac{1}{9}$

Step 9.2

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{1}{8} \cdot \frac{9}{9} - \frac{1}{9}$

Step 9.3

To write $- \frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{8} \cdot \frac{9}{9} - \frac{1}{9} \cdot \frac{8}{8}$

Step 9.4

Write each expression with a common denominator of $72$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.4.1

Multiply $\frac{1}{8}$ by $\frac{9}{9}$ .

$\frac{9}{8 \cdot 9} - \frac{1}{9} \cdot \frac{8}{8}$

Step 9.4.2

Multiply $8$ by $9$ .

$\frac{9}{72} - \frac{1}{9} \cdot \frac{8}{8}$

Step 9.4.3

Multiply $\frac{1}{9}$ by $\frac{8}{8}$ .

$\frac{9}{72} - \frac{8}{9 \cdot 8}$

Step 9.4.4

Multiply $9$ by $8$ .

$\frac{9}{72} - \frac{8}{72}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{9 - 8}{72}$

Step 9.6

Subtract $8$ from $9$ .

$\frac{1}{72}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{72}$

Decimal Form:

$0.013\overline{8}$