Calculus Examples

$lim_{x \to - \infty} \ln (| \frac{2}{x} - e^{4} |)$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the logarithm.

$\ln (lim_{x \to - \infty} | \frac{2}{x} - e^{4} |)$

Step 1.2

Move the limit inside the absolute value signs.

$\ln (| lim_{x \to - \infty} \frac{2}{x} - e^{4} |)$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- \infty$ .

$\ln (| lim_{x \to - \infty} \frac{2}{x} - lim_{x \to - \infty} e^{4} |)$

Step 1.4

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

$\ln (| 2 lim_{x \to - \infty} \frac{1}{x} - lim_{x \to - \infty} e^{4} |)$

Step 2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\ln (| 2 \cdot 0 - lim_{x \to - \infty} e^{4} |)$

Step 3

Evaluate the limit.

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Step 3.1

Evaluate the limit of $e^{4}$ which is constant as $x$ approaches $- \infty$ .

$\ln (| 2 \cdot 0 - e^{4} |)$

Step 3.2

Simplify the answer.

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Step 3.2.1

Multiply $2$ by $0$ .

$\ln (| 0 - e^{4} |)$

Step 3.2.2

Subtract $e^{4}$ from $0$ .

$\ln (| - e^{4} |)$

Step 3.2.3

$- e^{4}$ is approximately $- 54.59815003$ which is negative so negate $- e^{4}$ and remove the absolute value

$\ln (e^{4})$

Step 3.2.4

Use logarithm rules to move $4$ out of the exponent.

$4 \ln (e)$

Step 3.2.5

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

$4 \cdot 1$

Step 3.2.6

Multiply $4$ by $1$ .

$4$