I've encountered the following problem: Evaluate $\lim_{x\to 0^+}\cfrac{x^{x^x}}{x}.$
This is readily a "$\frac{0}{0}$" form, so I used L'Hopital's rule, but it got seriously messy, and fast. Can anyone recommend an alternative approach?
I've encountered the following problem: Evaluate $\lim_{x\to 0^+}\cfrac{x^{x^x}}{x}.$
This is readily a "$\frac{0}{0}$" form, so I used L'Hopital's rule, but it got seriously messy, and fast. Can anyone recommend an alternative approach?
You can avoid series by taking advantage of the known limit $\lim_{x\to 0^+}x^x=1$. Let $f(x)=\ln\left(\dfrac{x^{x^x}}x\right)=(x^x-1)\ln x$. Then
$\begin{align*} \lim_{x\to 0^+}f(x)&=\lim_{x\to 0^+}(x^x-1)\ln x\\\\ &=\lim_{x\to 0^+}\frac{\ln x}{\frac1{x^x-1}}\\\\ &=\lim_{x\to 0^+}\frac{1/x}{-\left(x^x-1\right)^{-2}x^x(1+\ln x)}\\\\ &=-\lim_{x\to 0^+}\frac{\left(x^x-1\right)^2}{x(1+\ln x)}\\\\ &=-\lim_{x\to 0^+}\frac{2\left(x^x-1\right)(1+\ln x)}{2+\ln x}\\\\ &=-2\left(\lim_{x\to 0^+}(x^x-1)\right)\left(\lim_{x\to 0^+}\frac{1+\ln x}{2+\ln x}\right)\\\\ &=-2\cdot0\cdot1\\\\&=0\;, \end{align*}$
and the desired limit is $1$.
All we need is a nice enough series expansion for $x^x$ about $0$, which can be obtained by rewriting $x^x$ as $\exp \left( x \log (x)\right)$ and looking at the Taylor series of $\exp \left( y\right)$. Now if we look at $f(x) = \log \left( \dfrac{x^{x^x}}{x} \right) = \log \left( x^{x^x}\right) - \log x = \log (x) \left(x^x - 1 \right) \\= \log (x) \left(-1 + \left(1 + x \log (x) + \dfrac{x^2\log^2(x)}{2!} + \dfrac{x^3\log^3(x)}{3!} + \dfrac{x^4\log^4(x)}{4!} + \cdots\right) \right)\\=x \log^2(x) \left( 1 + \dfrac{x\log(x)}{2!} + \dfrac{x^2\log^2(x)}{3!} + \dfrac{x^3\log^3(x)}{4!} + \cdots\right)$ Hence, the limit of $f(x)$ as $x \to 0$ is $0$. Hence, $\lim_{x \rightarrow 0} \dfrac{x^{x^x}}{x} = 1$
Using the standard limits $\lim_{x \to 0^+} x \, (\ln x)^a = 0$ (for $a>0$) and $\lim_{t \to 0} \frac{e^t-1}{t} = 1$ we find that $ \frac{x^{x^x}}{x} = \frac{e^{x^x \ln x}}{e^{\ln x}} = e^{(x^x-1) \ln x} = \exp((e^{x \ln x}-1) \ln x) = \exp\left( \frac{e^{x \ln x} - 1}{x \ln x} \cdot x \,(\ln x)^2 \right) \to $ $\to \exp(1 \cdot 0) = 1 $ as $x \to 0^+$.
This really isn't bad. We have
$\displaystyle \frac{x^{x^x}}{x} = \large e^{\log(x^{x^x}) - \log(x)} = e^{\log(x)(e^{x\log(x)}-1) } = e^{\log(x)^{e^{x\log(x)}-1}} = x^{e^{x\log(x) - 1}} $
and so the continuity of $\exp(x)$ ensures that
$\displaystyle \lim_{x \to 0^+} \frac{x^{x^x}}{x} = \lim_{x \to 0^+} \large x^{e^{x\log(x) - 1}} = x^{e^{\lim_{...} (x\log(x) -1)}} = x^0 $
and so the limit is 1, as promised.
For a fast/simple solution i'm going to resort to 2 elementary limits, namely $\lim_{x\to0} \frac{\ln(1+x)}{x}=1$
and $\lim_{x\to0^+} x^x=1$. Let's proceed with the proof:
$\lim_{x\to 0^+}\cfrac{x^{x^x}}{x}=\lim_{x\to 0^+}x^{x^x-1}=\lim_{x\to 0^+}x^{{\frac{x^x-1}{\ln x^x}}\cdot\ln x^x}=\lim_{x\to 0^+}x^{\ln x^x}=\lim_{x\to 0^+}e^{x {(\ln x)}^{2}}=1.$
Q.E.D.
If $L$ is the desired limit then \begin{align} \log L &= \log\left\{\lim_{x \to 0^{+}}\frac{x^{x^{x}}}{x}\right\}\notag\\ &= \lim_{x \to 0^{+}}\log x^{x^{x} - 1}\text{ (via continuity of log)}\notag\\ &= \lim_{x \to 0^{+}}(x^{x} - 1)\log x\notag\\ &= \lim_{x \to 0^{+}}\frac{e^{x\log x} - 1}{x\log x}\cdot x(\log x)^{2}\notag\\ &= \lim_{x \to 0^{+}}1\cdot x(\log x)^{2}\notag\\ &= \lim_{x \to 0^{+}}x(\log x)^{2}\notag\\ &= 0\notag \end{align} Hence $L = 1$. Here we have used the standard result $\lim_{x \to 0^{+}}x(\log x)^{n} = 0$ for positive integer $n$. This is easily proved by putting $x = 1/y$ and letting $y \to \infty$. Clearly we have $\lim_{x \to 0^{+}}x(\log x)^{n} = \lim_{y \to \infty}(-1)^{n}\cdot\frac{(\log y)^{n}}{y} = 0$