Calculus Examples

$lim_{x \to 8} 3 x - \frac{2}{x}$

Step 1

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

$lim_{x \to 8} 3 x - lim_{x \to 8} \frac{2}{x}$

Step 2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 8} x - lim_{x \to 8} \frac{2}{x}$

Step 3

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

$3 lim_{x \to 8} x - 2 lim_{x \to 8} \frac{1}{x}$

Step 4

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

$3 lim_{x \to 8} x - 2 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}$

Step 5

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

$3 lim_{x \to 8} x - 2 \frac{1}{lim_{x \to 8} x}$

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 $x$ .

$3 \cdot 8 - 2 \frac{1}{lim_{x \to 8} x}$

Step 6.2

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

$3 \cdot 8 - 2 (\frac{1}{8})$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $3$ by $8$ .

$24 - 2 (\frac{1}{8})$

Step 7.1.2

Cancel the common factor of $2$ .

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Step 7.1.2.1

Factor $2$ out of $- 2$ .

$24 + 2 (- 1) \frac{1}{8}$

Step 7.1.2.2

Factor $2$ out of $8$ .

$24 + 2 \cdot - 1 \frac{1}{2 \cdot 4}$

Step 7.1.2.3

Cancel the common factor.

$24 + 2 \cdot - 1 \frac{1}{2 \cdot 4}$

Step 7.1.2.4

Rewrite the expression.

$24 - \frac{1}{4}$

Step 7.2

To write $24$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$24 \cdot \frac{4}{4} - \frac{1}{4}$

Step 7.3

Combine $24$ and $\frac{4}{4}$ .

$\frac{24 \cdot 4}{4} - \frac{1}{4}$

Step 7.4

Combine the numerators over the common denominator.

$\frac{24 \cdot 4 - 1}{4}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Multiply $24$ by $4$ .

$\frac{96 - 1}{4}$

Step 7.5.2

Subtract $1$ from $96$ .

$\frac{95}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{95}{4}$

Decimal Form:

$23.75$