WEBVTT

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Here is the problem: (sqrt(x) minus 2)/(x minus 4).

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Find the limit of (sqrt(x) minus 2) / (x minus 4) as x → 4.

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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 4, so the limit equals its value there.

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So the answer is Limit equals 1 over 4 approximately 0.25.

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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(4.01) equals 0.24984395, f(4.001) equals 0.24998438, f(4.0001) equals 0.24999844, f(4.00001) equals 0.24999984, which checks out.

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Numerical values from the left agree: f(3.99) equals 0.25015645, f(3.999) equals 0.25001563, f(3.9999) equals 0.25000156, f(3.99999) equals 0.25000016, which checks out.