Friday, 24 November 2017

real analysis - Compute limlimitsxto0+efrac1sqrtxcdot(1sqrtx)frac1x

Evaluate lim
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?

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