Calculus Examples

$lim_{x \to 8} \frac{1}{x} - \frac{2 x}{x - 1}$

Step 1

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

$lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} \frac{2 x}{x - 1}$

Step 2

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

$\frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{2 x}{x - 1}$

Step 3

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

$\frac{1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{2 x}{x - 1}$

Step 4

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

$\frac{1}{lim_{x \to 8} x} - 2 lim_{x \to 8} \frac{x}{x - 1}$

Step 5

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

$\frac{1}{lim_{x \to 8} x} - 2 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1}$

Step 6

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

$\frac{1}{lim_{x \to 8} x} - 2 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - lim_{x \to 8} 1}$

Step 7

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

$\frac{1}{lim_{x \to 8} x} - 2 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1}$

Step 8

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

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Step 8.1

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

$\frac{1}{8} - 2 \frac{lim_{x \to 8} x}{lim_{x \to 8} x - 1 \cdot 1}$

Step 8.2

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

$\frac{1}{8} - 2 \frac{8}{lim_{x \to 8} x - 1 \cdot 1}$

Step 8.3

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

$\frac{1}{8} - 2 \frac{8}{8 - 1 \cdot 1}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Multiply $- 1$ by $1$ .

$\frac{1}{8} - 2 \frac{8}{8 - 1}$

Step 9.1.1.2

Subtract $1$ from $8$ .

$\frac{1}{8} - 2 (\frac{8}{7})$

Step 9.1.2

Multiply $- 2 (\frac{8}{7})$ .

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Step 9.1.2.1

Combine $- 2$ and $\frac{8}{7}$ .

$\frac{1}{8} + \frac{- 2 \cdot 8}{7}$

Step 9.1.2.2

Multiply $- 2$ by $8$ .

$\frac{1}{8} + \frac{- 16}{7}$

Step 9.1.3

Move the negative in front of the fraction.

$\frac{1}{8} - \frac{16}{7}$

Step 9.2

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{8} \cdot \frac{7}{7} - \frac{16}{7}$

Step 9.3

To write $- \frac{16}{7}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{8} \cdot \frac{7}{7} - \frac{16}{7} \cdot \frac{8}{8}$

Step 9.4

Write each expression with a common denominator of $56$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.4.1

Multiply $\frac{1}{8}$ by $\frac{7}{7}$ .

$\frac{7}{8 \cdot 7} - \frac{16}{7} \cdot \frac{8}{8}$

Step 9.4.2

Multiply $8$ by $7$ .

$\frac{7}{56} - \frac{16}{7} \cdot \frac{8}{8}$

Step 9.4.3

Multiply $\frac{16}{7}$ by $\frac{8}{8}$ .

$\frac{7}{56} - \frac{16 \cdot 8}{7 \cdot 8}$

Step 9.4.4

Multiply $7$ by $8$ .

$\frac{7}{56} - \frac{16 \cdot 8}{56}$

Step 9.5

Combine the numerators over the common denominator.

$\frac{7 - 16 \cdot 8}{56}$

Step 9.6

Simplify the numerator.

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Step 9.6.1

Multiply $- 16$ by $8$ .

$\frac{7 - 128}{56}$

Step 9.6.2

Subtract $128$ from $7$ .

$\frac{- 121}{56}$

Step 9.7

Move the negative in front of the fraction.

$- \frac{121}{56}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{121}{56}$

Decimal Form:

$- 2.16071428 \dots$