Application of squeeze theorem to $\displaystyle\lim_{x \rightarrow 0}x^4\cos(\frac{\pi}{x^2})+1$ Use squeeze theorem to evaluate
$$\lim_{x \rightarrow 0}x^4\cos(\frac{\pi}{x^2})+1$$
Can I sandwich it between $-x^4\cos(\frac{\pi}{x^2})+1$ and $x^4\cos(\frac{\pi}{x^2})+1$?

You can squeeze using $1-x^4$ and $1+x^4$. First, show $1-x^4 \leq 1+x^4cos\left(\frac{\pi}{x^2}\right) \leq 1+x^4$. Then we can use the squeeze theorem to show $\displaystyle\lim_{x\rightarrow 0} 1+x^4cos\left(\frac{\pi}{x^2}\right) = 1$ as $\displaystyle\lim_{x\rightarrow 0} 1 + x^4 = 1 = \displaystyle\lim_{x\rightarrow 0} 1 - x^4$.
