Trying to determine $\lim_{x\to\infty }\frac {x}{\sin x} $

Question

I am trying to determine $$\lim_{x\to \infty}\frac {x}{\sin x} $$

According to the definition of limit at infinity , I think that does not exist.

Am I right? If not please correct me.

Answer 1

You can see that by using the following two sequence:

1.) $x_n:= \frac{\pi}{2}+ 2\pi\cdot n$

2.) $y_n:= \frac{3\pi}{2}+ 2\pi\cdot n$

Inserting theses series into your expression gives two series, where the first diverges to $+\infty$ and the second to $-\infty$. This implies that the expression has no limit.

A similar agument shows that the limit $lim_\infty \frac{1}{sin(x)}$ does not exists, too.

Answer 2

You are right. The sine function keeps alternating between positive and negative values while the quotient becomes arbitrarily large in absolute value.
(Aside: $\lim_{x\to\infty} x/|\sin(x)| = +\infty$.)