Exercise 13

Consider the function $g(x) = x^{2x}$, which is defined for all $x \gt 0$. Observe that $\lim_{x \to 0^+} g(x)$ is indeterminate due to its form of $0^0$. (Think about how we know that $0^k = 0$ for all $k \gt 0$, while $b^0 = 1$ for all $b \ne 0$, but that neither rule can apply to $0^0$.)

1. Let $h(x) = \ln(g(x))$. Explain why $h(x) = 2x \ln(x)$.

2. Next, explain why it is equivalent to write $h(x) = \frac{2\ln(x)}{\frac{1}{x}}$.

3. Use L'Hôpital's Rule and your work in (b) to compute $\lim_{x \to 0^+} h(x)$.

4. Based on the value of $\lim_{x \to 0^+} h(x)$, determine $\lim_{x \to 0^+} g(x)$.