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Here is the problem: (x squared minus 9)/(x minus 3).

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Find the limit of (x squared minus 9) / (x minus 3) as x → 3.

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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 3, so the limit equals its value there.

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So the answer is Limit equals 6.

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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(3.01) equals 6.01, f(3.001) equals 6.001, f(3.0001) equals 6.0001, f(3.00001) equals 6.00001, which checks out.

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Numerical values from the left agree: f(2.99) equals 5.99, f(2.999) equals 5.999, f(2.9999) equals 5.9999, f(2.99999) equals 5.99999, which checks out.