Calculus Examples

$lim_{x \to 8} \frac{5}{x} - \frac{x}{3}$

Step 1

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

$lim_{x \to 8} \frac{5}{x} - lim_{x \to 8} \frac{x}{3}$

Step 2

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$5 lim_{x \to 8} \frac{1}{x} - lim_{x \to 8} \frac{x}{3}$

Step 3

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

$5 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{x}{3}$

Step 4

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

$5 \frac{1}{lim_{x \to 8} x} - lim_{x \to 8} \frac{x}{3}$

Step 5

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$5 \frac{1}{lim_{x \to 8} x} - \frac{1}{3} lim_{x \to 8} x$

Step 6

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

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Step 6.1

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

$5 (\frac{1}{8}) - \frac{1}{3} lim_{x \to 8} x$

Step 6.2

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

$5 (\frac{1}{8}) - \frac{1}{3} \cdot 8$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Combine $5$ and $\frac{1}{8}$ .

$\frac{5}{8} - \frac{1}{3} \cdot 8$

Step 7.1.2

Multiply $- \frac{1}{3} \cdot 8$ .

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Step 7.1.2.1

Multiply $8$ by $- 1$ .

$\frac{5}{8} - 8 (\frac{1}{3})$

Step 7.1.2.2

Combine $- 8$ and $\frac{1}{3}$ .

$\frac{5}{8} + \frac{- 8}{3}$

Step 7.1.3

Move the negative in front of the fraction.

$\frac{5}{8} - \frac{8}{3}$

Step 7.2

To write $\frac{5}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5}{8} \cdot \frac{3}{3} - \frac{8}{3}$

Step 7.3

To write $- \frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{5}{8} \cdot \frac{3}{3} - \frac{8}{3} \cdot \frac{8}{8}$

Step 7.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.4.1

Multiply $\frac{5}{8}$ by $\frac{3}{3}$ .

$\frac{5 \cdot 3}{8 \cdot 3} - \frac{8}{3} \cdot \frac{8}{8}$

Step 7.4.2

Multiply $8$ by $3$ .

$\frac{5 \cdot 3}{24} - \frac{8}{3} \cdot \frac{8}{8}$

Step 7.4.3

Multiply $\frac{8}{3}$ by $\frac{8}{8}$ .

$\frac{5 \cdot 3}{24} - \frac{8 \cdot 8}{3 \cdot 8}$

Step 7.4.4

Multiply $3$ by $8$ .

$\frac{5 \cdot 3}{24} - \frac{8 \cdot 8}{24}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{5 \cdot 3 - 8 \cdot 8}{24}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $5$ by $3$ .

$\frac{15 - 8 \cdot 8}{24}$

Step 7.6.2

Multiply $- 8$ by $8$ .

$\frac{15 - 64}{24}$

Step 7.6.3

Subtract $64$ from $15$ .

$\frac{- 49}{24}$

Step 7.7

Move the negative in front of the fraction.

$- \frac{49}{24}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{49}{24}$

Decimal Form:

$- 2.041\overline{6}$