To find the horizontal asymptote of the natural logarithm function, f(x) = ln(x), we need to consider the behavior of the function as x approaches infinity and as x approaches zero.
1. **As x approaches infinity**: The natural log function grows without bound, meaning that as x increases, ln(x) also increases. Mathematically, we can express this as:
lim (x → ∞) ln(x) = ∞
Since the function approaches infinity, there is no horizontal asymptote in this direction.
2. **As x approaches zero from the right**: The natural logarithm function decreases without bound. In this case, as x gets closer to zero, ln(x) approaches negative infinity:
lim (x → 0⁺) ln(x) = -∞
Again, this indicates that there is no horizontal asymptote at this end as well.
In summary, the natural logarithm function f(x) = ln(x) does not have any horizontal asymptotes since it approaches both positive and negative infinity in its respective intervals. Thus, the function grows indefinitely as x increases, and it declines indefinitely as x approaches zero from the right.