Value of an exponential function What is the value of the exponential function $y=e^{x^{x}-1}$ at $x=0$? I graphed the function on desmos and the value at $x=0$ is $1$. However, I do not know how to show this analytically.

This function is not defined at $0$, but you can compute the limit $\lim_{x\to0}e^{x^x-1}=e^{\lim_{x\to0}x^x-1}$. It happens that$$\lim_{x\to0}x^x=\lim_{x\to0}e^{x\log x}=e^{\lim_{x\to0}x\log x}$$and that\begin{align}\lim_{x\to0}x\log x&=\lim_{x\to0}\frac{\log x}{\frac1x}\\&=\lim_{x\to0}\frac{\frac1x}{-\frac1{x^2}}\\&=-\lim_{x\to0}x\\&=0.\end{align}So, $\lim_{x\to0}x^x=e^0=1$ and therefore $\lim_{x\to0}e^{x^x-1}=e^0=1$.
