Step by step solution to "What is the value of the exponential function..."

Trace Glass

Answered question

2022-10-15

What is the value of the exponential function

y = e^{x^{x} - 1}

at x=0?

Bridget Acevedo

Beginner2022-10-16Added 19 answers

This function is not defined at 0, but you can compute the limit

lim_{x \to 0} e^{x^{x} - 1} = e^{lim_{x \to 0} x^{x} - 1}

. It happens that

lim_{x \to 0} x^{x} = lim_{x \to 0} e^{x \log x} = e^{lim_{x \to 0} x \log x}

and that

\begin{aligned} lim_{x \to 0} x \log x & = lim_{x \to 0} \frac{\log x}{\frac{1}{x}} \\ = lim_{x \to 0} \frac{\frac{1}{x}}{- \frac{1}{x^{2}}} \\ = - lim_{x \to 0} x \\ = 0. \end{aligned}

So,

lim_{x \to 0} x^{x} = e^{0} = 1

and therefore

lim_{x \to 0} e^{x^{x} - 1} = e^{0} = 1

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