$Find $\lim _{x \rightarrow 0} x^{x}$$

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$Find $\lim _{x \rightarrow 0} x^{x}$$

How should this limit be computed?

$$\lim _{x \rightarrow 0} x^{x}$$

Calculus Precalculus Algebra

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Let $f(x)=x^{x}$. Then $\ln f(x)=\ln x^x = x \ln x.$ Hence
$$\lim_{x\rightarrow 0} \ln f(x)=\lim_{x\rightarrow 0}x\ln x=\lim_{x\rightarrow 0}\frac{\ln x}{\frac{1}{x}}=\lim_{x\rightarrow 0}\frac{\frac{1}{x}}{-\frac{1}{x^2}}=\lim_{x\rightarrow 0} -x=0,$$
where we have used the L'Hospital's Rule. Thus
$$\ln (\lim_{x\rightarrow 0}f(x) )=\lim_{x\rightarrow 0} \ln f(x)=0.$$
Hence
$$\lim_{x\rightarrow 0}x^x=\lim_{x\rightarrow 0}f(x) =1.$$

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$Find $\lim _{x \rightarrow 0} x^{x}$$

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