Final answer:
The limit of ln(x) as x approaches infinity is infinity. As x becomes larger, the natural logarithm of x also increases without bound, since ln(x) asks to what power e must be raised to achieve x. Therefore, answer C) [infinity] is correct.
Step-by-step explanation:
The question asks, "What is the limit as x approaches infinity of ln(x)?" The function ln(x), which stands for natural logarithm, is the inverse of the exponential function e^x. This means ln(e^x) = x and e^(ln x) = x. By understanding this relationship, we can determine behavior of the ln function as x grows larger. As x approaches infinity, the natural logarithm of x increases without bound because you are essentially asking to what power should the constant e (approximately 2.7182818) be raised to get increasingly large numbers. Consequently, ln(x) also approaches infinity as x approaches infinity.
Therefore, the answer to the question is C) [infinity] (infinity), as the natural logarithm function will continue to grow larger and does not have an upper limit. The properties of logarithms and the behavior of other functions with asymptotes or limits can help us understand this concept better. For example, the function y = 1/x has an asymptote at y = 0, because as x approaches infinity, y approaches zero, similarly as x approaches zero from the positive side, y approaches infinity. However, the behavior of the ln function is such that as x gets larger, ln(x) also gets larger, and not closer to a finite number.
It is worth noting that the approach of the logarithm function towards infinity is not abrupt but gradual and unbounded, which makes sense if we recognize logarithmic growth as the inverse of exponential growth — just as exponential functions grow rapidly, logarithmic functions grow slowly but also unboundedly. The significance of the natural logarithm and exponential function relationship in this context underscores how they help us compute various mathematical and scientific problems, particularly those involving growth and decay.