Question on using sandwich rule with trig and abs function to show that a limit exists.

Question

Past paper Question:

For the following function, determine whether $\lim_{x\to\infty}f(x)$ exists, and compute the limit if it exists. Justify your answers.

$$f(x)= \dfrac{\sin(x)+1}{\left| x \right|}$$

Attempt:

Consider the fact that $-1 \le \sin(x) \le 1$ (for all $x$), which implies $0 \le \sin(x) +1\le 2$. Dividing by $\left| x \right|,$

$$\color{green}{ \frac{0}{\left| x \right|}} \le \color{blue}{ \frac{\sin( x)+1}{\left| x \right|}} \le \color{red}{ \frac{2}{\left| x \right|}}$$

Since green tens to $0$, and the red tends to $0$, (via AOL for $\dfrac{1}{x}$ as $x \rightarrow \infty)$, blue will tend to $0$ via the algebra of limits and sandwich theorem, is this correct, or will the absolute value of $x$ effect this?

Answer 1

Well done.

Note that when $x\to \infty$ implies $|x| \to \infty$