Final answer:
No single value of b can satisfy both limits for the function f(x) = bx as x approaches infinity. For the limit to be zero, b must be between 0 and 1, and for the limit to be infinity, b must be greater than 1.
Step-by-step explanation:
The question is asking for the values of b for which two specific limits of the function f(x) = bx are obtained: the limit as x approaches infinity where f(x) approaches zero, and the limit as x approaches infinity where f(x) approaches infinity. To solve this, we consider the properties of exponential functions.
For f(x) = bx to satisfy lim f(x) = 0 as x → ∞, b must be a number between 0 and 1 (0 < b < 1). This is because as x increases without bound, an exponential function with a base that is a fraction will approach zero.
Conversely, for f(x) = bx to satisfy lim f(x) = ∞ as x → ∞, b must be greater than 1 (b > 1). When the base of an exponential function is greater than 1, the function's value will increase without bound as x increases.
In summary, no single value of b can satisfy both conditions simultaneously. The values of b that satisfy each condition are mutually exclusive.