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Here is the problem: (5x cubed minus x)/(2x cubed plus 7).

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Find the limit of (5 x cubed minus x) / (2 x cubed plus 7) as x → ∞.

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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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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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Simplify.

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The expression is constant.

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So the answer is Limit equals 5 over 2 approximately 2.5.

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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 2.4999648, f(1271.8281) equals 2.4999997, f(12718.281) equals 2.5, f(127182.81) equals 2.5, which checks out.