WEBVTT

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Here is the problem: x sin(1/x).

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Find the limit of x sin(1 / x) as x → ∞.

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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 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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Substitute x equals 1/t (t → 0 to the power plus ).

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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 as x grows agree: f(127.18281) equals 0.9999897, f(1271.8281) equals 0.9999999, f(12718.281) equals 1, f(127182.81) equals 1, which checks out.