Evaluate $\lim_{x\to 0} (\frac {(1+x)^{\frac 1x}}{e})^{\frac{1}{\sin x}}$ Can I get a starting hint to solve this? The second index of $\frac{1}{\sin x}$ is throwing me off and I really don’t know what to do. Thanks!
Call this $L$, then evaluate $\ell:=\ln L$ viz.$$\ell=\lim_{x\to0}\frac{\tfrac1x\ln(1+x)-1}{\sin x}.$$Various methods show the numerator (denominator) is asymptotic to $-\tfrac12x$ ($x$), so $\ell=-\tfrac12$ i.e. $L=\tfrac{1}{\sqrt{e}}$.
Edit: as requested, here's an approach not using the Maclaurin series of $\ln(1+x)$. By L'Hôpital's rule,$$\frac{\ln(1+x)-x}{x\sin x}\sim\frac{1/(1+x)-1}{x\cos x+\sin x}=\frac{-1/(1+x)}{\cos x+\tfrac{\sin x}{x}}\to\frac{-1}{1+1}=-\tfrac12.$$
I did the following within my CAS:

Then I did

Limit[((1 + x)^(1/x)/E)^(1/x), x -&gt; 0]

(*   1/Sqrt[Exp1] *)
Limit[((1 + x)^(1/x)/E)^(x/6), x -&gt; 0]
1

Make use of the rule that if a limit is a product of two limits than the limit of the product exists and is the product of the limits. Calculation rules with limits in general. The rest is even more 1 for small x approaching 0.
N@(1/Sqrt[E])
0.606531

Table[((1 + x)^(1/x)/E)^(1/Sin[x]), {x, 0.0001, 0.0003, 0.0001}]
{0.606551, 0.606571, 0.606591}

ListLinePlot@
 Table[{x, ((1 + x)^(1/x)/E)^(1/Sin[x])}, {x, 0.0001, 0.0003, 0.0001}]


For small x this has an affine representation.
Series[((1 + x)^(1/x)/E)^(1/Sin[x]), {x, 0, 2}]
SeriesData[x, 0, {
 E^Rational[-1, 2], Rational[1, 3] E^Rational[-1, 2], 
  Rational[-5, 18] E^Rational[-1, 2]}, 0, 3, 1]