Calculus Examples

$lim_{x \to 8} {(1 + \frac{4}{x^{5}})}^{x}$

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

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

Step 1.2

Combine the numerators over the common denominator.

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(\frac{x^{5} + 4}{x^{5}})}^{x}$ as $e^{\ln ({(\frac{x^{5} + 4}{x^{5}})}^{x})}$ .

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

Step 2.2

Expand $\ln ({(\frac{x^{5} + 4}{x^{5}})}^{x})$ by moving $x$ outside the logarithm.

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

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

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

Step 3.2

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

$e^{lim_{x \to 8} x \cdot lim_{x \to 8} \ln (\frac{x^{5} + 4}{x^{5}})}$

Step 3.3

Move the limit inside the logarithm.

$e^{lim_{x \to 8} x \cdot \ln (lim_{x \to 8} \frac{x^{5} + 4}{x^{5}})}$

Step 3.4

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

$e^{lim_{x \to 8} x \cdot \ln (\frac{lim_{x \to 8} x^{5} + 4}{lim_{x \to 8} x^{5}})}$

Step 3.5

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

$e^{lim_{x \to 8} x \cdot \ln (\frac{lim_{x \to 8} x^{5} + lim_{x \to 8} 4}{lim_{x \to 8} x^{5}})}$

Step 3.6

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$e^{lim_{x \to 8} x \cdot \ln (\frac{{(lim_{x \to 8} x)}^{5} + lim_{x \to 8} 4}{lim_{x \to 8} x^{5}})}$

Step 3.7

Evaluate the limit of $4$ which is constant as $x$ approaches $8$ .

$e^{lim_{x \to 8} x \cdot \ln (\frac{{(lim_{x \to 8} x)}^{5} + 4}{lim_{x \to 8} x^{5}})}$

Step 3.8

Move the exponent $5$ from $x^{5}$ outside the limit using the Limits Power Rule.

$e^{lim_{x \to 8} x \cdot \ln (\frac{{(lim_{x \to 8} x)}^{5} + 4}{{(lim_{x \to 8} x)}^{5}})}$

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 $x$ .

$e^{8 \cdot \ln (\frac{{(lim_{x \to 8} x)}^{5} + 4}{{(lim_{x \to 8} x)}^{5}})}$

Step 4.2

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

$e^{8 \cdot \ln (\frac{8^{5} + 4}{{(lim_{x \to 8} x)}^{5}})}$

Step 4.3

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

$e^{8 \cdot \ln (\frac{8^{5} + 4}{8^{5}})}$

Step 5

Simplify the answer.

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Step 5.1

Simplify $8 \cdot \ln (\frac{8^{5} + 4}{8^{5}})$ by moving $8$ inside the logarithm.

$e^{\ln ({(\frac{8^{5} + 4}{8^{5}})}^{8})}$

Step 5.2

Exponentiation and log are inverse functions.

${(\frac{8^{5} + 4}{8^{5}})}^{8}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Raise $8$ to the power of $5$ .

${(\frac{32768 + 4}{8^{5}})}^{8}$

Step 5.3.2

Add $32768$ and $4$ .

${(\frac{32772}{8^{5}})}^{8}$

Step 5.4

Raise $8$ to the power of $5$ .

${(\frac{32772}{32768})}^{8}$

Step 5.5

Cancel the common factor of $32772$ and $32768$ .

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Step 5.5.1

Factor $4$ out of $32772$ .

${(\frac{4 (8193)}{32768})}^{8}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $4$ out of $32768$ .

${(\frac{4 \cdot 8193}{4 \cdot 8192})}^{8}$

Step 5.5.2.2

Cancel the common factor.

${(\frac{4 \cdot 8193}{4 \cdot 8192})}^{8}$

Step 5.5.2.3

Rewrite the expression.

${(\frac{8193}{8192})}^{8}$

Step 5.6

Apply the product rule to $\frac{8193}{8192}$ .

$\frac{8193^{8}}{8192^{8}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{8193^{8}}{8192^{8}}$

Decimal Form:

$1.00097697 \dots$