Question

select the correct answer.
what is the equation of the asymptote for this function?
f(x) = ln x + 5
a. x = 0
b. x = -5
c. y = 0
d. y = -5

Explanation:

Step1: Recall the domain of ln(x)

The natural logarithm function \( \ln(x) \) is defined for \( x > 0 \), and as \( x \) approaches 0 from the right (\( x \to 0^+ \)), \( \ln(x) \) approaches \( -\infty \).

Step2: Analyze the vertical asymptote of \( f(x)=\ln x + 5 \)

For the function \( f(x)=\ln x + 5 \), the vertical asymptote occurs where the function behaves in a way that it approaches \( \pm\infty \). Since the \( \ln(x) \) part drives the behavior near \( x = 0 \), the vertical asymptote of \( f(x)=\ln x + 5 \) is the same as the vertical asymptote of \( \ln(x) \), which is \( x = 0 \). Horizontal asymptotes: As \( x \to \infty \), \( \ln(x) \to \infty \), so there is no horizontal asymptote (eliminating options C and D). The vertical asymptote is not \( x=-5 \) because \( \ln(x) \) is not defined for \( x \leq 0 \), so the function doesn't approach \( \pm\infty \) near \( x=-5 \) (eliminating option B).

Answer:

A. \( x = 0 \)