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B5

Part of 1954 Putnam

Problems(1)

Putnam 1954 B5

Source: Putnam 1954

7/17/2022
Let f(x)f(x) be a real-valued function, defined for −1<x<1-1<x<1 for which f′(0)f'(0) exists. Let (an),(bn)(a_n) , (b_n) be two sequences such that −1<an<0<bn<1-1 <a_n <0 <b_n <1 for all nn and lim⁡n→∞an=0=lim⁡n→∞bn.\lim_{n \to \infty } a_n = 0 =\lim_{n \to \infty} b_n. Prove that lim⁡n→∞f(bn)−f(an)bn−an=f′(0). \lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).
Putnamfunctionlimitdifferentiability