Let \(f\) be any function with the following properties:

• \(f\) is twice continuously differentiable in a neighborhood of zero¹;
• \(f(0) = 0\);
• \(f'(0) = 0\);
• \(f''(0) = 1\).

Assume \(t\) is a constant and compute

\[ \lim_{x\to\infty} xf\left(\frac{t}{\sqrt{x}}\right) . \]

Continuity?

A full credit solution must clearly explain how and why continuity is being used along the way.

Footnotes

1. This means that \(f\) and its first two derivatives are all continuous functions at and around zero.