To determine the equation of the asymptote for the function [tex]\( f(x) = \ln x + 5 \)[/tex], let's analyze the properties of the function step-by-step.
The function [tex]\( f(x) = \ln x \)[/tex] represents the natural logarithm of [tex]\( x \)[/tex]. The key characteristics of the natural logarithm function are as follows:
1. The domain of [tex]\( \ln x \)[/tex] is [tex]\( x > 0 \)[/tex], as the natural logarithm is only defined for positive [tex]\( x \)[/tex].
2. As [tex]\( x \)[/tex] approaches 0 from the positive side ([tex]\( x \to 0^+ \)[/tex]), the value of [tex]\( \ln x \)[/tex] decreases without bound, approaching negative infinity. This behavior indicates a vertical asymptote at [tex]\( x = 0 \)[/tex].
Next, let's consider the effect of the transformation applied to [tex]\( \ln x \)[/tex]:
1. Adding 5 to [tex]\( \ln x \)[/tex] (i.e., [tex]\( f(x) = \ln x + 5 \)[/tex]) shifts the graph of [tex]\( \ln x \)[/tex] vertically upward by 5 units.
Despite this vertical shift, the vertical asymptote does not change because the asymptote is dependent on the behavior of [tex]\( \ln x \)[/tex] as [tex]\( x \)[/tex] approaches 0. The addition of a constant does not affect the asymptote's location.
From this analysis, it's clear that the function [tex]\( f(x) = \ln x + 5 \)[/tex] retains the vertical asymptote at [tex]\( x = 0 \)[/tex], just like the base function [tex]\( \ln x \)[/tex].
Therefore, the correct answer is:
A. [tex]\( x = 0 \)[/tex]
