Question

Which of the following could describe a single logarithmic function f?

(В
lim f(x) = -∞ and lim f(x) =
==
x→0+
818
lim f(x) = -∞o and lim f(x) = k, where k is a positive constant
x-0+
x-x
lim f(x) = ∞ and lim f(x) = 0
20+
*→∞
lim f(x) = ∞ and lim f(x) = -∞
2-0+
€48

Answer 1

The answer is D.

Step-by-step explanation:

Logarithmic functions act with no bound as x --> infinity. Therefore, our answer cannot be B or C. Answer choice A represents a quadratic function, as the line must bend to form a parabola. This does not follow logarithmic behavior. So, the only choice that follows logarithmic behavior is D.

Answer 2

The behavior of a logarithmic function as x approaches zero from the positive side is to head towards negative infinity, while as x approaches infinity, the logarithmic function's value grows towards positive infinity. The correct description for a logarithmic function is that the limit as x approaches 0+ is negative infinity, and as x approaches infinity, the limit is infinity.

Step-by-step explanation:

The four options given describe the behavior of a function f(x) as x approaches certain values. The focus of the question seems to be on identifying which of these options could correctly describe the behavior of a logarithmic function f(x).

We know that a logarithmic function would indeed head towards negative infinity as x approaches zero from the positive side (x→0+), because the logarithm of a number very close to zero is a large negative number. Additionally, as x approaches infinity, the logarithm grows without bounds, albeit at a decreasing rate, which means the second limit would be infinity (limx→∞ f(x) = ∞).

Therefore, the correct option that can describe a logarithmic function f is the third one: limx→0+ f(x) = -∞ and limx→∞ f(x) = ∞. This is consistent with the known properties of logarithmic functions such as the natural logarithm ln(x), which showcase similar limits with asymptotes.