I want to find the following limit $$\lim_{x \to 0^+}e^{-ax\sqrt{2^{b+c/x}-1}}.$$ where $a,b,c$ are positive constants.
My Attempt:
I can use the series formula for exponential to have following form $$\lim_{x\to 0^+} \left[1+ax\sqrt{2^{b+c/x}-1}+.5\left(ax\sqrt{2^{b+c/x}-1}\right)^2+(3!)^{-1}\left(ax\sqrt{2^{b+c/x}-1}\right)^3+.....\right]^{-1}$$ but unfortunately I do not even know how to solve the very second term in this series. I need your help. Thanks in advance.
Assuming $x \to 0^+$. Then roughly speaking, as $y = c/x \to \infty$, the $b$ and $-1$ become insignificant:
$$e^{-ax\sqrt{2^{b+c/x}-1}} \sim \exp\left(-ax \sqrt{2^{c/x}}\right) = \exp\left(-ac \frac{\sqrt{2^{y}}}{y}\right).$$
Note that $\sqrt{2^{y}} = 2^{y/2}$ grows much faster than $y$ (since it's exponential) and hence
$$\lim_{y\to \infty} \frac{\sqrt{2^{y}}}{y} = \infty.$$ Therefore,
$$\lim_{y\to\infty}\exp\left(-ac \frac{\sqrt{2^{y}}}{y}\right) = e^{-\infty} = 0 = \lim_{x\to0^+}e^{-ax\sqrt{2^{b+c/x}-1}}.$$
