# 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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