Answer
a) As $x$ approaches $0$, $x^x$ approaches $1$.
b) The graph heads for the value $1$ on the $y$-axis.
Work Step by Step
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.
