grandes-ecoles 2012 QIII.D

grandes-ecoles · France · centrale-maths1__psi Sequences and Series Limit Evaluation Involving Sequences
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Does $Lf(x) - \dfrac{1}{2x^2} + \dfrac{1}{x}$ admit a finite limit at $0^+$? 
In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).

Does $Lf(x) - \dfrac{1}{2x^2} + \dfrac{1}{x}$ admit a finite limit at $0^+$?
Paper Questions
QI.A QI.B QI.C QII.A QII.B QII.C QIII.A QIII.B QIII.C QIII.D QIV.A QIV.B QIV.C QIV.D QV.A QV.B QV.C QV.D QV.E QV.F QV.G QVI.A QVI.B QVI.C QVII.A QVII.B QVIII.A QVIII.B QVIII.C QVIII.D QVIII.E QVIII.F QVIII.G