How to prove $ \lim_{x\rightarrow 0^+} \frac{1}{\sqrt{x}} \cos(1/x) $ does not exist

Question

How do I prove that this limit does not exist?

$$ \lim_{x\rightarrow 0^+} \frac{1}{\sqrt{x}} \cos(1/x) $$

When plugging in $0^+$ in $x$ we get $-\infty \times \cos(\infty)$ which is undefined. however I am trying to prove that it doesn't exist and am not capable? Do I use the squeeze rule or toast rule? I have tried a few different ways and haven't managed anything

Answer 1

As for any $\delta$, $0<x<\delta\implies \dfrac 1x$ is unbounded,

$$\max_{0<x<\delta}\left|\frac1{\sqrt x}\cos\dfrac1x\right|=\max_{0<x<\delta}\frac1{\sqrt x},$$ which is obviously unbounded.

Answer 2

we can write $$f(x)=\frac{1}{\sqrt{x}} \cos\frac{1}{x}$$ Take $$x_n=\frac{1}{2n\pi}, x'_n= \frac{1}{(2n+1)\pi}$$ $f(x_n)=(2\pi n)^{1/2}\cos (2\pi n)>0$ and $f(x'_n)=((2n+1)\pi)^{1/2} \cos((2n+1)\pi)<0$, these two values are essentially unequal. the $\lim_{x \to 0} f(x)$ does not exist.

Answer 3

$$\lim_{x\to 0^+} \frac{\cos(x^{-1})}{\sqrt{x}}=\lim_{n\to + \infty} \frac{\cos((\frac{1}{n\pi})^{-1})}{\sqrt{\frac{1}{n\pi}}}=\sqrt{\pi} \lim_{n \to +\infty} (-1)^n \sqrt{n}$$ And that limit clearly does not exist, so initial limit doesn't exist.