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Here is the problem: (x cubed minus 8)/(x minus 2).

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Find the limit of (x cubed minus 8) / (x minus 2) as x → 2.

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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 2, so the limit equals its value there.

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So the answer is Limit equals 12.

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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(2.01) equals 12.0601, f(2.001) equals 12.006001, f(2.0001) equals 12.0006, f(2.00001) equals 12.00006, which checks out.

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Numerical values from the left agree: f(1.99) equals 11.9401, f(1.999) equals 11.994001, f(1.9999) equals 11.9994, f(1.99999) equals 11.99994, which checks out.