University Calculus: Early Transcendentals (3rd Edition)

a) As $x$ approaches $0$, $x^x$ approaches $1$. b) The graph heads for the value $1$ on the $y$-axis.
a) Using the calculator to calculate $x^x$ for $x=0.1, 0.01, 0.001$ and so on: - For $x=0.1$: $x^x\approx0.794328$ - For $x=0.01$: $x^x\approx0.954993$ - For $x=0.001$: $x^x\approx0.993116$ - For $x=0.0001$: $x^x\approx0.999079$ - For $x=0.00001$: $x^x\approx0.999885$ - For $x=0.000001$: $x^x\approx0.999986$ - For $x=0.0000001$: $x^x\approx0.999998$ We can easily see the pattern here is that as $x$ gets smaller and approaches $0$, $x^x$ approaches $1$. b) The graph of the function $y=x^x$ is included below. Looking at the graph, it seems to head for the value $1$ on the $y$-axis.