Question 47
Let $f : [1, +\infty) \to (0, +\infty)$ be a continuous function having the following properties:
(i) The function $g : [1, +\infty) \to (0, +\infty)$ given by $g(x) = \frac{f(x)}{x}$ has limit at $+\infty$,
(ii) The function $h : [1, +\infty) \to (0, +\infty)$ given by $h(x) = \frac{1}{x} \int_{1}^{x} f(t) dt$ has finite limit at $+\infty$.
a) Show that $\lim_{x \to +\infty} g(x) = 0$.
b) Show that $\lim_{x \to +\infty} \frac{1}{x^2} \int_1^x f^2(t) dt = 0$.
(i) The function $g : [1, +\infty) \to (0, +\infty)$ given by $g(x) = \frac{f(x)}{x}$ has limit at $+\infty$,
(ii) The function $h : [1, +\infty) \to (0, +\infty)$ given by $h(x) = \frac{1}{x} \int_{1}^{x} f(t) dt$ has finite limit at $+\infty$.
a) Show that $\lim_{x \to +\infty} g(x) = 0$.
b) Show that $\lim_{x \to +\infty} \frac{1}{x^2} \int_1^x f^2(t) dt = 0$.