Wednesday, 17 August 2016

real analysis - Finding limxrightarrow0+fracxx1x



I'm trying to solve the limit lim



I think we should use L'Hospital rule and the limit becomes



\lim_{x\rightarrow 0^+} -x^{-x}(\log x + 1)=\lim_{x\rightarrow 0^+} \frac{\log x + 1}{-x^{x}}= +\infty




Is it right?



I've tried to modify the form and not use L'Hospital's rule but without success.


Answer



Without the Hospital's rule: \frac{x^{-x}-1}{x}=-\frac{x^x-1}{x\cdot x^x}=-\frac{e^{x\ln{x}}-1}{x\ln{x}}\cdot\ln{x}\cdot\frac{1}{x^x}\rightarrow+\infty.


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