Final answer:
The vertical asymptote for the function f(x) = 1/x is at x = 0. To the left of it, the function approaches negative infinity, and to the right, it approaches positive infinity. This behavior characterizes the hyperbolic nature of the function near the asymptote.
Step-by-step explanation:
To find the vertical asymptote of a function, we look for values of x where the function could become undefined or tends to infinity. For the given function f(x) = 1/x, the vertical asymptote is at x = 0, because as x approaches 0, the value of the function grows without bound.
Describing the behavior of f(x), as x approaches 0 from the left (negative values), f(x) goes to negative infinity, indicating that the curve is heading downwards towards the vertical asymptote. As x approaches 0 from the right (positive values), f(x) goes to positive infinity, meaning the curve is going upwards towards the vertical asymptote.
Graphing the Asymptotic Behavior
When graphing the function, we should label it and use appropriate scales. Since the only vertical asymptote is at x = 0, to the left the function will descend steeply, while to the right it will rise sharply, never touching the y-axis but moving closer to it indefinitely. These are key characteristics of a hyperbolic function.