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Here is the problem: sqrt(x squared plus x) minus x.

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Find the limit of sqrt(x squared plus x) minus x as x → ∞.

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Indeterminate form ∞ minus ∞ — rewrite. Multiply by the conjugate: (u minus v)(u plus v)/(u plus v) equals (u squared minus v squared)/(u plus v).

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Indeterminate form ∞/∞ — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule).

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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 0.5, which equals 1 over 2 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 1 over 2 approximately 0.5.

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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 as x grows agree: f(127.18281) equals 0.49902101, f(1271.8281) equals 0.49990175, f(12718.281) equals 0.49999017, f(127182.81) equals 0.49999902, which checks out.