Y by
Suppose that
is twice continuously differentiable on
, and satisfies the conditions:
Prove that:
![\[
\lim_{x \to +\infty} x f'(x) = 0.
\]](//latex.artofproblemsolving.com/4/d/3/4d3487c0be2069b8e1a44e7fe88b8d3b4456557d.png)


![\[
\lim_{x \to +\infty} x f(x) = 0 \quad \text{and} \quad \lim_{x \to +\infty} x f''(x) = 0.
\]](http://latex.artofproblemsolving.com/3/e/5/3e536d8b5357b75e68d24c12b5cb0202edd52c8c.png)
![\[
\lim_{x \to +\infty} x f'(x) = 0.
\]](http://latex.artofproblemsolving.com/4/d/3/4d3487c0be2069b8e1a44e7fe88b8d3b4456557d.png)
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