real analysis - Compute $limlimits_{xto 0_+}e^{frac{1}{sqrt{x}}}cdot(1-sqrt{x})^frac{1}{x}$

Evaluate $$\lim_{x\to 0_+}e^{\frac{1}{\sqrt{x}}}\cdot(1-\sqrt{x})^\frac{1}{x}$$
I tried using $$\lim_{x\to 0}(1+x)^\frac{1}{x} = e$$ like so:
$$l = \lim_{x\to 0_+}e^\frac{1}{\sqrt{x}}\cdot\bigg[\big(1+(-\sqrt{x})\big)^{-\frac{1}{\sqrt{x}}}\bigg]^{\frac{-1}{\sqrt{x}}} = \lim_{x\to 0_+}e^{\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x}}} = e^0 = 1$$
However, the right answer is $\frac{1}{\sqrt e}$. Why is it that the whole expression in square brackets can't be taken as $e$ in this case?