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Answer from Sia

Posted 5 months ago

Solution by Steps

step 1

Evaluate the limit of $\frac{\ln(x)}{x}$ as $x$ approaches $0$ from the positive side

step 2

Recognize that as $x$ approaches $0^+$, $\ln(x)$ approaches $-\infty$ and $x$ approaches $0$

step 3

Consider the behavior of the function: the numerator goes to $-\infty$ faster than the denominator goes to $0$

step 4

Conclude that the limit $\lim_{x \to 0^+} \frac{\ln(x)}{x}$ is $-\infty$

Answer

$-\infty$

Key Concept

Limits involving natural logarithms as the argument approaches zero

Explanation

As $x$ approaches $0$ from the positive side, the natural logarithm of $x$ becomes very large in the negative direction much faster than $x$ approaches zero, causing the fraction to approach negative infinity.

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