Solving the limit that involves exponential and power

Question

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.

Answer 1

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,

$$\lim_{y\to\infty}\exp\left(-ac \frac{\sqrt{2^{y}}}{y}\right) = e^{-\infty} = 0 = \lim_{x\to0^+}e^{-ax\sqrt{2^{b+c/x}-1}}.$$

Answer 2

Hint: Try to show that $x\sqrt {2^{b + c/x} -1}\to \infty$ as $x\to 0^+.$