Question

`lim_(x->oo) sin(2x)/x` help me?

Answer 1

`lim_{x \to \infty}\frac{\sin(2x)}{x} = 0`

Explanation:

The sine function has no limit as `x\to\infty`, since it keeps going up and down between `-1` and `1`. Actually, this is the key point: in absolute value, this function never exceeds `1`.

On the other hand, the denominator keeps growing as much as we want, since `x \to \infty`. In formulas, we have

`|\sin(2x)|\le 1 \iff \frac{-1}{x} \le \frac{\sin(2x)}{x} \le \frac{1}{x}`

If we consider the limit as `x \to \infty` to all sides, we have

`0 \le \frac{\sin(2x)}{x} \le 0`

and thus our quantity is squeezed between two other quantities which tend to zero, and must tend to zero itself.