Need of L-Hospital's Rule

Calculus Level pending

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:

  1. \(\displaystyle \lim_{x \to 0^+} f(x)\)
  2. \(\displaystyle \lim_{x \to 0^+} x^x\)
  3. \(\displaystyle \lim_{x \to 0^+} e^{\log x^x}\)
  4. \(\displaystyle \lim_{x \to 0^+} e^{x\log x}\)
  5. \(e^{\lim_{x \to 0^+}(x \log x)}\)
  6. \(e^{\lim_{x \to 0^+} \frac d{dx}(x \log x)}\)
  7. \(e^{\lim_{x \to 0^+} (1+\log x)}\)
  8. \(e^\infty \)
  9. \(\infty \) (Hence proved)
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