Calculate the limit $L=\lim_{x\to 0^+}\left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x$.

Question

Question: Calculate the limit $$L=\lim_{x\to 0^+}\left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x.$$

I'm thinking of using infinitesimal, but I'm not used to these kind of analysis arguments. Can someone explain how to deal with these kind of problems? Thanks in advance.

Answer 1

Note that

$$ \left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x=e^{x\log \left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right) }\to1$$

indeed

$$x\log \left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)\to0 $$

since

$$x\log \left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)=x \left[\log \sqrt{x} +\log\left(2\frac{\sin\left(\sqrt{x}\right)}{\sqrt{x}}+\sin\left(\frac{1}{x}\right)\right)\right]=\sqrt{x} \left[ \sqrt{x} \log \sqrt{x} + \sqrt{x} \log\left(2\frac{\sin\left(\sqrt{x}\right)}{\sqrt{x}}+\sin\left(\frac{1}{x}\right)\right)\right]\to0\cdot(0+0)=0$$

Answer 2

If the $\sin(1/x)$ is giving you trouble, bound it:

$$ \left(\sqrt{x} - \frac{1}{3}x^{3/2}\right)^x \leq \left(2\sin\left(\sqrt{x}\right)+\sqrt{x}\sin\left(\frac{1}{x}\right)\right)^x \leq \left( 3\sqrt{x}\right)^x. $$