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lim frac{f^{(2n)}1}{(2n)!}, f(x)=\frac{e^x}{x}

Source: LIMIT 2020 Cat 2 Objective P16

June 9, 2020
limitderivativecalculus

Problem Statement

The nthn^{th} derivative of a function f(x)f(x) (if it exists) is denoted by f(n)(x)f^{(n)}(x) . Let f(x)=exxf(x)=\frac{e^x}{x}. Suppose ff is differentiable infinitely many times in (0,∞)(0,\infty) . Then find lim⁡n→∞f(2n)1(2n)!\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}
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