To determine the equation of the asymptote for the function [tex]\( f(x) = \ln x + 5 \)[/tex], let's analyze the properties of the components of this function.
1. Understanding the Natural Logarithm Function [tex]\( \ln x \)[/tex]:
- The natural logarithm function, [tex]\( \ln x \)[/tex], is defined for [tex]\( x > 0 \)[/tex].
- As [tex]\( x \)[/tex] approaches zero from the positive side (i.e., [tex]\( x \to 0^+ \)[/tex]), [tex]\( \ln x \)[/tex] decreases without bound, meaning it goes to negative infinity.
2. Vertical Asymptotes:
- A vertical asymptote is a vertical line [tex]\( x = a \)[/tex] where the function [tex]\( f(x) \)[/tex] increases or decreases without bound as [tex]\( x \)[/tex] approaches [tex]\( a \)[/tex].
- For the function [tex]\( \ln x \)[/tex], there is a vertical asymptote at [tex]\( x = 0 \)[/tex] because [tex]\( \ln x \)[/tex] is undefined for [tex]\( x \leq 0 \)[/tex] and approaches negative infinity as [tex]\( x \)[/tex] gets closer to zero from the right.
3. Adding a Constant:
- The function [tex]\( f(x) = \ln x + 5 \)[/tex] is simply a vertical shift of the function [tex]\( \ln x \)[/tex] by 5 units upward. This vertical shift does not affect the location of the vertical asymptote.
- Therefore, the vertical asymptote remains at [tex]\( x = 0 \)[/tex].
Having gone through the detailed analysis, we can conclude that the equation of the asymptote for the function [tex]\( f(x) = \ln x + 5 \)[/tex] is:
[tex]\[ \boxed{x = 0} \][/tex]
Hence, the correct answer is:
A. [tex]\( x = 0 \)[/tex]
