Calculus Examples

$lim_{x \to 0} \ln ({(x^{4} + 8 x + 1)}^{x - 1})$

Step 1

Move the limit inside the logarithm.

$\ln (lim_{x \to 0} {(x^{4} + 8 x + 1)}^{x - 1})$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${(x^{4} + 8 x + 1)}^{x - 1}$ as $e^{\ln ({(x^{4} + 8 x + 1)}^{x - 1})}$ .

$\ln (lim_{x \to 0} e^{\ln ({(x^{4} + 8 x + 1)}^{x - 1})})$

Step 2.2

Expand $\ln ({(x^{4} + 8 x + 1)}^{x - 1})$ by moving $x - 1$ outside the logarithm.

$\ln (lim_{x \to 0} e^{(x - 1) \ln (x^{4} + 8 x + 1)})$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$\ln (e^{lim_{x \to 0} (x - 1) \ln (x^{4} + 8 x + 1)})$

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\ln (e^{lim_{x \to 0} x - 1 \cdot lim_{x \to 0} \ln (x^{4} + 8 x + 1)})$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\ln (e^{(lim_{x \to 0} x - lim_{x \to 0} 1) \cdot lim_{x \to 0} \ln (x^{4} + 8 x + 1)})$

Step 3.4

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot lim_{x \to 0} \ln (x^{4} + 8 x + 1)})$

Step 3.5

Move the limit inside the logarithm.

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot \ln (lim_{x \to 0} x^{4} + 8 x + 1)})$

Step 3.6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot \ln (lim_{x \to 0} x^{4} + lim_{x \to 0} 8 x + lim_{x \to 0} 1)})$

Step 3.7

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot \ln ({(lim_{x \to 0} x)}^{4} + lim_{x \to 0} 8 x + lim_{x \to 0} 1)})$

Step 3.8

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot \ln ({(lim_{x \to 0} x)}^{4} + 8 lim_{x \to 0} x + lim_{x \to 0} 1)})$

Step 3.9

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\ln (e^{(lim_{x \to 0} x - 1 \cdot 1) \cdot \ln ({(lim_{x \to 0} x)}^{4} + 8 lim_{x \to 0} x + 1)})$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\ln (e^{(0 - 1 \cdot 1) \cdot \ln ({(lim_{x \to 0} x)}^{4} + 8 lim_{x \to 0} x + 1)})$

Step 4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\ln (e^{(0 - 1 \cdot 1) \cdot \ln (0^{4} + 8 lim_{x \to 0} x + 1)})$

Step 4.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\ln (e^{(0 - 1 \cdot 1) \cdot \ln (0^{4} + 8 \cdot 0 + 1)})$

Step 5

Simplify the answer.

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Step 5.1

Use logarithm rules to move $(0 - 1 \cdot 1) \cdot \ln (0^{4} + 8 \cdot 0 + 1)$ out of the exponent.

$(0 - 1 \cdot 1) \cdot \ln (0^{4} + 8 \cdot 0 + 1) \ln (e)$

Step 5.2

Multiply $- 1$ by $1$ .

$(0 - 1) \cdot \ln (0^{4} + 8 \cdot 0 + 1) \ln (e)$

Step 5.3

Subtract $1$ from $0$ .

$- 1 \cdot \ln (0^{4} + 8 \cdot 0 + 1) \ln (e)$

Step 5.4

Simplify each term.

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Step 5.4.1

Raising $0$ to any positive power yields $0$ .

$- 1 \cdot \ln (0 + 8 \cdot 0 + 1) \ln (e)$

Step 5.4.2

Multiply $8$ by $0$ .

$- 1 \cdot \ln (0 + 0 + 1) \ln (e)$

Step 5.5

Add $0$ and $0$ .

$- 1 \cdot \ln (0 + 1) \ln (e)$

Step 5.6

Add $0$ and $1$ .

$- 1 \cdot \ln (1) \ln (e)$

Step 5.7

The natural logarithm of $1$ is $0$ .

$- 1 \cdot 0 \ln (e)$

Step 5.8

Multiply $- 1$ by $0$ .

$0 \ln (e)$

Step 5.9

The natural logarithm of $e$ is $1$ .

$0 \cdot 1$

Step 5.10

Multiply $0$ by $1$ .

$0$