Answer :
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.
1. Understanding the Natural Logarithm Function:
- The natural logarithm function [tex]\( \ln x \)[/tex] is only defined for [tex]\( x > 0 \)[/tex].
- As [tex]\( x \)[/tex] approaches 0 from the positive side (i.e., [tex]\( x \to 0^+ \)[/tex]), [tex]\( \ln x \)[/tex] tends to [tex]\(-\infty \)[/tex].
- The graph of [tex]\( \ln x \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex], because it becomes infinitely large in the negative direction as [tex]\( x \)[/tex] gets very close to 0.
2. Effect of Adding a Constant:
- The function [tex]\( f(x) = \ln x + 5 \)[/tex] can be seen as a vertical shift of the base function [tex]\( \ln x \)[/tex] by 5 units upward.
- While this vertical shift affects the value of the function, it does not impact the location of the vertical asymptote, because shifts along the vertical axis do not move the position where [tex]\( x \)[/tex] leads [tex]\( \ln x \)[/tex] to [tex]\(-\infty \)[/tex].
3. Identifying the Asymptote:
- Since the addition of 5 only shifts the graph up, the vertical asymptote of the function [tex]\( f(x) = \ln x + 5 \)[/tex] remains at the same place as the asymptote of [tex]\( \ln x \)[/tex].
- Therefore, the vertical asymptote for this function is at [tex]\( x = 0 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{x = 0} \][/tex]
Thus:
The correct option is A. [tex]\( x = 0 \)[/tex].
1. Understanding the Natural Logarithm Function:
- The natural logarithm function [tex]\( \ln x \)[/tex] is only defined for [tex]\( x > 0 \)[/tex].
- As [tex]\( x \)[/tex] approaches 0 from the positive side (i.e., [tex]\( x \to 0^+ \)[/tex]), [tex]\( \ln x \)[/tex] tends to [tex]\(-\infty \)[/tex].
- The graph of [tex]\( \ln x \)[/tex] has a vertical asymptote at [tex]\( x = 0 \)[/tex], because it becomes infinitely large in the negative direction as [tex]\( x \)[/tex] gets very close to 0.
2. Effect of Adding a Constant:
- The function [tex]\( f(x) = \ln x + 5 \)[/tex] can be seen as a vertical shift of the base function [tex]\( \ln x \)[/tex] by 5 units upward.
- While this vertical shift affects the value of the function, it does not impact the location of the vertical asymptote, because shifts along the vertical axis do not move the position where [tex]\( x \)[/tex] leads [tex]\( \ln x \)[/tex] to [tex]\(-\infty \)[/tex].
3. Identifying the Asymptote:
- Since the addition of 5 only shifts the graph up, the vertical asymptote of the function [tex]\( f(x) = \ln x + 5 \)[/tex] remains at the same place as the asymptote of [tex]\( \ln x \)[/tex].
- Therefore, the vertical asymptote for this function is at [tex]\( x = 0 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{x = 0} \][/tex]
Thus:
The correct option is A. [tex]\( x = 0 \)[/tex].
