Calculus Examples

$lim_{x \to 8} \frac{4}{9 + \frac{1}{x}}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to 8} \frac{1}{9 + \frac{1}{x}}$

Step 1.2

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

$4 \frac{lim_{x \to 8} 1}{lim_{x \to 8} 9 + \frac{1}{x}}$

Step 1.3

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

$4 \frac{1}{lim_{x \to 8} 9 + \frac{1}{x}}$

Step 1.4

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

$4 \frac{1}{lim_{x \to 8} 9 + lim_{x \to 8} \frac{1}{x}}$

Step 1.5

Evaluate the limit of $9$ which is constant as $x$ approaches $8$ .

$4 \frac{1}{9 + lim_{x \to 8} \frac{1}{x}}$

Step 1.6

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

$4 \frac{1}{9 + \frac{lim_{x \to 8} 1}{lim_{x \to 8} x}}$

Step 1.7

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

$4 \frac{1}{9 + \frac{1}{lim_{x \to 8} x}}$

Step 2

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

$4 \frac{1}{9 + \frac{1}{8}}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the denominator.

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Step 3.1.1

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

$4 \frac{1}{9 \cdot \frac{8}{8} + \frac{1}{8}}$

Step 3.1.2

Combine $9$ and $\frac{8}{8}$ .

$4 \frac{1}{\frac{9 \cdot 8}{8} + \frac{1}{8}}$

Step 3.1.3

Combine the numerators over the common denominator.

$4 \frac{1}{\frac{9 \cdot 8 + 1}{8}}$

Step 3.1.4

Simplify the numerator.

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Step 3.1.4.1

Multiply $9$ by $8$ .

$4 \frac{1}{\frac{72 + 1}{8}}$

Step 3.1.4.2

Add $72$ and $1$ .

$4 \frac{1}{\frac{73}{8}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$4 (1 (\frac{8}{73}))$

Step 3.3

Multiply $\frac{8}{73}$ by $1$ .

$4 (\frac{8}{73})$

Step 3.4

Multiply $4 (\frac{8}{73})$ .

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Step 3.4.1

Combine $4$ and $\frac{8}{73}$ .

$\frac{4 \cdot 8}{73}$

Step 3.4.2

Multiply $4$ by $8$ .

$\frac{32}{73}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{32}{73}$

Decimal Form:

$0.43835616 \dots$