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Here is the problem: x to the power 4e to the power negative x.

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Find the limit of x to the power 4 e to the power negative x as x → ∞.

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Indeterminate form 0 times ∞ — rewrite as a quotient. Moving one factor to the denominator as its reciprocal turns 0 times ∞ into 0 over 0 or ∞/∞.

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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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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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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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Bounded numerator over an unbounded denominator. The numerator approaches 24 and the denominator grows without bound, so the quotient approaches 0.

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So the answer is Limit equals 0.

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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, f(1271.8281) equals 0, f(12718.281) equals 0, f(127182.81) equals 0, which checks out.