WEBVTT

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Here is the problem: x ln(x).

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Find the limit of x ln(x) as x → 0.

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Indeterminate form 0 times ∞ — rewrite as a quotient. Moving one factor to the denominator as its reciprocal turns 0 times ∞ into 0 over 0 or ∞/∞.

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Indeterminate form ∞/∞ — 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 0, so the limit equals its value there.

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So the answer is Limit equals 0.

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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: f(0.01) equals negative 0.046051702, f(0.001) equals negative 0.0069077553, f(0.0001) equals negative 0.00092103404, f(0.00001) equals negative 0.00011512925 (no clear trend at this precision).

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Numerical values from the left: f(negative 0.01) equals undefined, f(negative 0.001) equals undefined, f(negative 0.0001) equals undefined, f(negative 0.00001) equals undefined (no clear trend at this precision).