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Here is the problem: (x squared plus 1)/(x plus 1).

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Find the limit of (x squared plus 1) / (x plus 1) 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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Simplify.

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The expression grows without bound. As x → ∞, the values increase in size without limit.

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Conclusion. The function is unbounded, so the limit is +∞ (strictly, the limit does not exist as a real number; it diverges to +∞).

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So the answer is Limit equals ∞.

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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 126.19841, f(1271.8281) equals 1270.8297, f(12718.281) equals 12717.281, f(127182.81) equals 127181.81, which checks out.