MathDB

4

Part of 2018 VJIMC

Problems(2)

Possible limits of a function with a certain equation

Source: VJIMC 2018, Category I, Problem 4

4/14/2018
Determine all possible (finite or infinite) values of lim⁡x→−∞f(x)−lim⁡x→∞f(x),\lim_{x \to -\infty} f(x)-\lim_{x \to \infty} f(x), if f:R→Rf:\mathbb{R} \to \mathbb{R} is a strictly decreasing continuous function satisfying f(f(x))4−f(f(x))+f(x)=1f(f(x))^4-f(f(x))+f(x)=1 for all x∈Rx \in \mathbb{R}.
functionfunctional equationDecreasinglimit
Double integral with exponentials

Source: VJIMC 2018, Category II, Problem 4

4/14/2018
Compute the integral ∬R2(1−e−xyxy)2e−x2−y2dxdy.\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2} dx dy.
calculusintegrationDouble integral