Problem

Compare \(f(x)~=~\ln(x)\) and \(g(x)~=~1/x\) at the vertical asymptote \(x~=~0\). State the relevant domains and one-sided limits, and identify any horizontal asymptote.

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Both have vertical asymptote location x = 0, but ln(x) is defined only to the right and tends to ∞ as x → 0+. For 1/x, the left limit at 0 is −∞ and right limit is ∞; it also has horizontal asymptote y = 0, unlike ln(x).

Both have vertical asymptote location x = 0, but ln(x) is defined only to the right and tends to −∞ as x → 0+. For 1/x, the left limit at 0 is −∞ and right limit is ∞; it also has horizontal asymptote y = 0, unlike ln(x).

Both have vertical asymptote location x = 0, but ln(x) is defined only to the right and tends to −∞ as x → 0+. For 1/x, the left limit at 0 is ∞ and right limit is −∞; it also has horizontal asymptote y = 0, unlike ln(x).

Both have vertical asymptote location x = 0, but ln(x) is defined only to the right and tends to −∞ as x → 0+. For 1/x, the left limit at 0 is −∞ and right limit is ∞; ln(x) also has horizontal asymptote y = 0, unlike 1/x.

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Compare \(f(x)~=~\ln(x)\) and \(g(x)~=~1/x\) at the vertical asymptote \(x~=~0\). State the relevant domains and one-sided limits, and identify any horizontal asymptote.

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