How to show that $\lim_{x\to 0} \frac{x}{\sin\left(\frac{1}{x}\right)+x}$ doesn't exist

Question

Is there a simple proof, without using the epsilon/delta limit definition to prove that this limit when $x$ goes to $0$ doesn't exist?

$$ \lim_{x \to 0}\frac{x}{\sin\left(\frac{1}{x}\right)+x} $$

Answer 1

Take the following sequences: $$x_n=\frac{1}{2\pi n},\quad y_n=\frac{1}{\frac{\pi}{2}+2\pi n}.$$ Then both sequences tend to $0$ as $n\to\infty$. Note that $$\frac{x_n}{\sin\frac{1}{x_n}+x_n}=\frac{x_n}{\sin(2\pi n)+x_n}=\frac{x_n}{x_n}=1,$$ $$\frac{y_n}{\sin\frac{1}{y_n}+y_n}=\frac{y_n}{\sin(\frac{\pi}{2}+2\pi n)+y_n}=\frac{y_n}{1+y_n}\to 0,$$ Thus the limit doesn't exist.