1. **Problem Statement:** Test the limit, continuity, and differentiability of the function $$f(x) = \begin{cases} \cos \frac{1}{x} & x \neq 0 \\ 0 & x = 0 \end{cases}$$ at the point $x=0$. 2. **Limit:** We check $\lim_{x \to 0} f(x) = \lim_{x \to 0} \cos \frac{1}{x}$. Since $\cos \frac{1}{x}$ oscillates between $-1$ and $1$ infinitely often as $x \to 0$, the limit does not exist. 3. **Continuity:** For continuity at $x=0$, the limit must equal $f(0)$. Since the limit does not exist, $f$ is not continuous at $x=0$. 4. **Differentiability:** Differentiability implies continuity, so since $f$ is not continuous at $0$, it is not differentiable there. **Final conclusion:** $f$ is neither continuous nor differentiable at $x=0$.