Local vs Global Behavior Math Example 2

Solution

1. 1
   Local at $x=0$: Using the Taylor expansion $\sin(x)=x-\frac{x^3}{6}+\cdots$, we get $\frac{\sin x}{x}=1-\frac{x^2}{6}+\cdots\approx1$ near $x=0$. Locally the function is nearly constant at $1$.
2. 2
   Global for large $|x|$: $|\sin(x)|\leq1$ while $|x|\to\infty$, so $\left|\frac{\sin x}{x}\right|\leq\frac{1}{|x|}\to0$. The function oscillates with decreasing amplitude, approaching $0$.
3. 3
   This is an oscillatory decay: constant locally at $x=0$, oscillating and shrinking globally.