WEBVTT

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Here is the problem: (x squared minus 3x plus 2)/(x minus 1).

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Find the limit of (x squared minus 3 x plus 2) / (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 negative 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 negative 0.99, f(1.001) equals negative 0.999, f(1.0001) equals negative 0.9999, f(1.00001) equals negative 0.99999, which checks out.

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Numerical values from the left agree: f(0.99) equals negative 1.01, f(0.999) equals negative 1.001, f(0.9999) equals negative 1.0001, f(0.99999) equals negative 1.00001, which checks out.