Answer
a) As $x$ approaches $\infty$, the value of $f(x)$ stays the same at about $0.367879$.
b) The graph is a straight line from $x=0\to\infty$, and the $y$-value stays constantly at about $0.367879$.
Work Step by Step
a) Using the calculator to calculate $(\frac{1}{x})^{1/\ln x}$ for $x=10, 100, 1000$ and so on:
- For $x=10$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
- For $x=100$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
- For $x=1000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
- For $x=10000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
- For $x=100000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
- For $x=1000000$: $(\frac{1}{x})^{1/\ln x}\approx0.367879$
We can easily see the pattern here is that as $x$ gets larger and approaches $\infty$, the value of $(\frac{1}{x})^{1/\ln x}$ stays the same at approximately $0.367879$.
b) The graph of the function $f(x)=(\frac{1}{x})^{1/\ln x}$ is included below.
The graph is a straight line from $x=0\to\infty$, and the $y$-value stays constantly at about $0.367879$.
