With the following flawed work out I strived to prove that

\[\lim_{x \to 0} x^x= \infty\]

Which of the following step is made firstly incorrect?

Let a function \(f(x) = x^x \), where \(x>0\).

**Step**:

- \(\displaystyle \lim_{x \to 0^+} f(x)\)
- \(\displaystyle \lim_{x \to 0^+} x^x\)
- \(\displaystyle \lim_{x \to 0^+} e^{\log x^x}\)
- \(\displaystyle \lim_{x \to 0^+} e^{x\log x}\)
- \(e^{\lim_{x \to 0^+}(x \log x)}\)
- \(e^{\lim_{x \to 0^+} \frac d{dx}(x \log x)}\)
- \(e^{\lim_{x \to 0^+} (1+\log x)}\)
- \(e^\infty \)
- \(\infty \) (Hence proved)

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