Calculus Examples

$lim_{x \to 8} {(\frac{18 x}{18 x + 4})}^{5 x}$

Step 1

Cancel the common factor of $18$ and $18 x + 4$ .

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Step 1.1

Factor $2$ out of $18 x$ .

$lim_{x \to 8} {(\frac{2 (9 x)}{18 x + 4})}^{5 x}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $18 x$ .

$lim_{x \to 8} {(\frac{2 (9 x)}{2 (9 x) + 4})}^{5 x}$

Step 1.2.2

Factor $2$ out of $4$ .

$lim_{x \to 8} {(\frac{2 (9 x)}{2 (9 x) + 2 (2)})}^{5 x}$

Step 1.2.3

Factor $2$ out of $2 (9 x) + 2 (2)$ .

$lim_{x \to 8} {(\frac{2 (9 x)}{2 (9 x + 2)})}^{5 x}$

Step 1.2.4

Cancel the common factor.

$lim_{x \to 8} {(\frac{2 (9 x)}{2 (9 x + 2)})}^{5 x}$

Step 1.2.5

Rewrite the expression.

$lim_{x \to 8} {(\frac{9 x}{9 x + 2})}^{5 x}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(\frac{9 x}{9 x + 2})}^{5 x}$ as $e^{\ln ({(\frac{9 x}{9 x + 2})}^{5 x})}$ .

$lim_{x \to 8} e^{\ln ({(\frac{9 x}{9 x + 2})}^{5 x})}$

Step 2.2

Expand $\ln ({(\frac{9 x}{9 x + 2})}^{5 x})$ by moving $5 x$ outside the logarithm.

$lim_{x \to 8} e^{5 x \ln (\frac{9 x}{9 x + 2})}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$e^{lim_{x \to 8} 5 x \ln (\frac{9 x}{9 x + 2})}$

Step 3.2

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

$e^{5 lim_{x \to 8} x \ln (\frac{9 x}{9 x + 2})}$

Step 3.3

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

$e^{5 lim_{x \to 8} x \cdot lim_{x \to 8} \ln (\frac{9 x}{9 x + 2})}$

Step 3.4

Move the limit inside the logarithm.

$e^{5 lim_{x \to 8} x \cdot \ln (lim_{x \to 8} \frac{9 x}{9 x + 2})}$

Step 3.5

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

$e^{5 lim_{x \to 8} x \cdot \ln (9 lim_{x \to 8} \frac{x}{9 x + 2})}$

Step 3.6

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

$e^{5 lim_{x \to 8} x \cdot \ln (9 \frac{lim_{x \to 8} x}{lim_{x \to 8} 9 x + 2})}$

Step 3.7

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

$e^{5 lim_{x \to 8} x \cdot \ln (9 \frac{lim_{x \to 8} x}{lim_{x \to 8} 9 x + lim_{x \to 8} 2})}$

Step 3.8

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

$e^{5 lim_{x \to 8} x \cdot \ln (9 \frac{lim_{x \to 8} x}{9 lim_{x \to 8} x + lim_{x \to 8} 2})}$

Step 3.9

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$e^{5 lim_{x \to 8} x \cdot \ln (9 \frac{lim_{x \to 8} x}{9 lim_{x \to 8} x + 2})}$

Step 4

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

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Step 4.1

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

$e^{5 \cdot 8 \cdot \ln (9 \frac{lim_{x \to 8} x}{9 lim_{x \to 8} x + 2})}$

Step 4.2

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

$e^{5 \cdot 8 \cdot \ln (9 \frac{8}{9 lim_{x \to 8} x + 2})}$

Step 4.3

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

$e^{5 \cdot 8 \cdot \ln (9 \frac{8}{9 \cdot 8 + 2})}$

Step 5

Simplify the answer.

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Step 5.1

Multiply $5$ by $8$ .

$e^{40 \cdot \ln (9 \frac{8}{9 \cdot 8 + 2})}$

Step 5.2

Simplify $40 \cdot \ln (9 \frac{8}{9 \cdot 8 + 2})$ by moving $40$ inside the logarithm.

$e^{\ln ({(9 \frac{8}{9 \cdot 8 + 2})}^{40})}$

Step 5.3

Exponentiation and log are inverse functions.

${(9 \frac{8}{9 \cdot 8 + 2})}^{40}$

Step 5.4

Cancel the common factor of $8$ and $9 \cdot 8 + 2$ .

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Step 5.4.1

Factor $2$ out of $8$ .

${(9 \frac{2 \cdot 4}{9 \cdot 8 + 2})}^{40}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2$ out of $9 \cdot 8$ .

${(9 \frac{2 \cdot 4}{2 (9 \cdot 4) + 2})}^{40}$

Step 5.4.2.2

Factor $2$ out of $2$ .

${(9 \frac{2 \cdot 4}{2 (9 \cdot 4) + 2 (1)})}^{40}$

Step 5.4.2.3

Factor $2$ out of $2 (9 \cdot 4) + 2 (1)$ .

${(9 \frac{2 \cdot 4}{2 (9 \cdot 4 + 1)})}^{40}$

Step 5.4.2.4

Cancel the common factor.

${(9 \frac{2 \cdot 4}{2 (9 \cdot 4 + 1)})}^{40}$

Step 5.4.2.5

Rewrite the expression.

${(9 \frac{4}{9 \cdot 4 + 1})}^{40}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $9$ by $4$ .

${(9 \frac{4}{36 + 1})}^{40}$

Step 5.5.2

Add $36$ and $1$ .

${(9 (\frac{4}{37}))}^{40}$

Step 5.6

Multiply $9 (\frac{4}{37})$ .

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Step 5.6.1

Combine $9$ and $\frac{4}{37}$ .

${(\frac{9 \cdot 4}{37})}^{40}$

Step 5.6.2

Multiply $9$ by $4$ .

${(\frac{36}{37})}^{40}$

Step 5.7

Apply the product rule to $\frac{36}{37}$ .

$\frac{36^{40}}{37^{40}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{36^{40}}{37^{40}}$

Decimal Form:

$0.33421894 \dots$