WEBVTT

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

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Find the limit of x squared sin(1 / x) as x → 0.

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Evaluate the limiting behaviour. The dominant behaviour of each piece determines the limit.

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Identify the value. The rewritten expression settles to 0, which equals 0 to within 10 to the power minus 6; the converging numerical values below confirm it.

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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 agree: f(0.01) equals negative 0.000050636564, f(0.001) equals 8.268795e minus 7, f(0.0001) equals negative 3.056144e minus 9, f(0.00001) equals 3.57488e minus 12, which checks out.

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Numerical values from the left agree: f(negative 0.01) equals 0.000050636564, f(negative 0.001) equals negative 8.268795e minus 7, f(negative 0.0001) equals 3.056144e minus 9, f(negative 0.00001) equals negative 3.57488e minus 12, which checks out.