Let $A$ be a $2 \times 2$ orthogonal matrix such that $\operatorname{det}(A) = -1$. Show that $A$ represents reflection about a line in $\mathbb{R}^2$.
Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Determine (with appropriate justification) the following limit: $$\lim_{n \longrightarrow \infty} \int_0^1 nx^n f(x)\,\mathrm{d}x$$