Compute $\lim_{x\to0^+}⁡ \sqrt{x\left[\cos^2\left(\frac1{x^3}\right)-3\right]}$ using Squeeze theorem

Question

This is for a first year calc course, and we are using the squeeze theorem to determine the limits of composite functions containing trig functions. I'm familiar with the examples posted in other questions on this website, but haven't seen anything addressing the squeeze theorem as applied to trig functions squared.

My question is how to use the squeeze theorem on the example below:

$$\lim_{x\to0^+}⁡ \sqrt{x\left[\cos^2\left(\frac1{x^3}\right)-3\right]}$$

Normally, I would set the upper and lower bounds of this as being $-1 ≤ f(x) ≤ 1$ because that is the range of the cosine function,
but I wasn't sure how the $\cos^2$ would affect this.
Any help would be greatly appreciated!

Answer 1

hint

For $x\ne 0,$

$$0\le \cos^2(\frac{1}{x^3})\le 1$$

but your function is not defined on the right of $0$. you cannot compute $\lim_{x\to0^+}f(x)$.