Calculus Examples

$lim_{x \to 8} \frac{1}{x} - \frac{9}{x + 93}$

Step 1

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

$lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} \frac{9}{x + 93}$

Step 2

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

$\frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{9}{x + 93}$

Step 3

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

$\frac{1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{9}{x + 93}$

Step 4

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{lim_{x \to 8} x} - 9 lim_{x \to 8} \frac{1}{x + 93}$

Step 5

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

$\frac{1}{lim_{x \to 8} x} - 9 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x + 93}$

Step 6

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

$\frac{1}{lim_{x \to 8} x} - 9 \frac{1}{lim_{x \to 8} x + 93}$

Step 7

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

$\frac{1}{lim_{x \to 8} x} - 9 \frac{1}{lim_{x \to 8} x + lim_{x \to 8} 93}$

Step 8

Evaluate the limit of $93$ which is constant as $x$ approaches $8$ .

$\frac{1}{lim_{x \to 8} x} - 9 \frac{1}{lim_{x \to 8} x + 93}$

Step 9

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

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Step 9.1

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

$\frac{1}{8} - 9 \frac{1}{lim_{x \to 8} x + 93}$

Step 9.2

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

$\frac{1}{8} - 9 \frac{1}{8 + 93}$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Add $8$ and $93$ .

$\frac{1}{8} - 9 (\frac{1}{101})$

Step 10.1.2

Combine $- 9$ and $\frac{1}{101}$ .

$\frac{1}{8} + \frac{- 9}{101}$

Step 10.1.3

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{9}{101}$

Step 10.2

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{101}{101}$ .

$\frac{1}{8} \cdot \frac{101}{101} - \frac{9}{101}$

Step 10.3

To write $- \frac{9}{101}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{8} \cdot \frac{101}{101} - \frac{9}{101} \cdot \frac{8}{8}$

Step 10.4

Write each expression with a common denominator of $808$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.4.1

Multiply $\frac{1}{8}$ by $\frac{101}{101}$ .

$\frac{101}{8 \cdot 101} - \frac{9}{101} \cdot \frac{8}{8}$

Step 10.4.2

Multiply $8$ by $101$ .

$\frac{101}{808} - \frac{9}{101} \cdot \frac{8}{8}$

Step 10.4.3

Multiply $\frac{9}{101}$ by $\frac{8}{8}$ .

$\frac{101}{808} - \frac{9 \cdot 8}{101 \cdot 8}$

Step 10.4.4

Multiply $101$ by $8$ .

$\frac{101}{808} - \frac{9 \cdot 8}{808}$

Step 10.5

Combine the numerators over the common denominator.

$\frac{101 - 9 \cdot 8}{808}$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $- 9$ by $8$ .

$\frac{101 - 72}{808}$

Step 10.6.2

Subtract $72$ from $101$ .

$\frac{29}{808}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{29}{808}$

Decimal Form:

$0.035\overline{8910}$