WEBVTT

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Here is the problem: ln(x)/(x minus 1).

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Find the limit of ln(x) / (x minus 1) as x → 1.

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Indeterminate form 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule).

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Simplify.

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Direct substitution. The expression is continuous at x equals 1, so the limit equals its value there.

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So the answer is Limit equals 1.

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A limit describes the value a function approaches, not necessarily the value it takes. When substitution gives an indeterminate form, the expression is rewritten (L prime Hôpital prime s rule, a conjugate, or logarithms) until substitution works.

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Numerical values from the right agree: f(1.01) equals 0.99503309, f(1.001) equals 0.99950033, f(1.0001) equals 0.99995, f(1.00001) equals 0.999995, which checks out.

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Numerical values from the left agree: f(0.99) equals 1.0050336, f(0.999) equals 1.0005003, f(0.9999) equals 1.00005, f(0.99999) equals 1.000005, which checks out.