Problem 38 - Calculus: Early Transcendental Functions

Question

Starting with \(\quad e^{x}=\lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}, \quad\) show \(\quad\) that \(\ln x=\lim _{n \rightarrow \infty}\left[n\left(x^{1 / n}-1\right)\right] .\) Assume that if \(\lim _{n \rightarrow \infty} x_{n}=x,\) then \(\lim _{n \rightarrow \infty}\left[n\left(x_{n}^{1 / n}-1\right)\right]=\lim _{n \rightarrow \infty}\left[n\left(x^{1 / n}-1\right)\right].\)

Step-by-Step Solution

Answer

By taking the logarithm of both sides in the limit definition of the exponential function and some simplification, we finally get the limit definition of logarithm.

Step 1: Declaring the Equation

We start by writing the equation to be proved, which is \( \ln x=\lim_{n \rightarrow \infty}\left[n\left(x^{1 / n}-1\right)\right] \).

Step 2: Doing the substitution

We then replace \(x\) with \(\left(1+\frac{x}{n}\right)^{n}\) based on the definition of the exponential function. The equation now becomes: \( \ln \left(1+\frac{x}{n}\right)^{n} = \lim_{n \rightarrow \infty}\left[n\left(\left(\left(1+\frac{x}{n}\right)^{n}\right)^{1 / n}-1\right)\right] \).

Step 3: Simplifying the Equation

After replacement, by simplifying the equation, we get \(n\ln \left(1+\frac{x}{n}\right)= \lim_{n \rightarrow \infty}\left[n\left(\left(1+\frac{x}{n}\right) - 1\right)\right]\). If we further simplify this equation, we get \(n\ln \left(1+\frac{x}{n}\right)= \lim_{n \rightarrow \infty}\left[\frac{x}{n}\right] = x \).

Step 4: Final Result

We then can say that \( \ln x = \lim_{n \rightarrow \infty}n\ln \left(1+\frac{x}{n}\right) \). A closer look reveals that this is exactly the limit definition of logarithm.

Key Concepts

Exponential Function

Imagine the exponential function as a mathematical escalator, rapidly taking values ever higher as you go along. It is formally written as e^x, where e is Euler's number, approximately equal to 2.71828. This isn't just any number; it's the base rate of growth shared by all continually growing processes. It's as natural to the universe of mathematics as pi is to circles.

The power of the exponential function really shines through when it comes to compounding growth or decay. It describes processes that increase or decrease very quickly. When you’re looking at a graph of an exponential function, expect a curve that starts off deceptively mild and then skyrockets when x gets larger. This function is incredibly important across various fields, making understanding it a valuable part of your mathematical toolbox.

Natural Logarithm

The natural logarithm, or ln, is the flip side of the exponential function. It's like the detective function of mathematics, figuring out what power you need to raise e to in order to get a particular number.

Formally, if you have e^x = y, then ln(y) = x. It’s as if you're asking, 'How many times do I need to multiply e by itself to reach y?'. The natural logarithm answers that question. It's 'natural' because it's based on the natural growth constant e, and it shows up in countless areas, especially where the rate of change is related to the current size—think interest in economics or populations in biology.

Limits in Calculus

Limits are the heartbeat of calculus, giving us a precise language for describing the behavior of functions as they get close to a certain point. They tell us where a function is headed, which might not be where it currently is, like a directional sign on the road of numbers.

In essence, when mathematicians say \( \text{lim}_{x \to c} f(x) = L \), they're saying that as x gets infinitely close to c, the function f(x) approaches a value L. It's about predicting the destination without actually reaching per se, and it's a cornerstone for tasks like finding the slope of a curve (derivatives) or the area under a curve (integrals). Limits help us make sense of things that are otherwise indescribable, like infinity.

Proofs in Mathematics

Proofs are the glue that holds the vast, intricate structure of mathematics together. They’re the 'aha!' moments which transform assumptions into facts. A proof is a logical argument that demonstrates the truth of a mathematical statement beyond any doubt.

Think of them as rigorous explanations that show a statement holds water in every single possible scenario without exception. They can be direct, going straight from A to B, or they can be indirect, assuming the opposite of what you want to prove and showing a contradiction. There are also proofs by induction, where you prove a basis case and then show that if it's true for one case, it's true for the next, and so on. Math without proofs would be like buildings without foundations—eventually, everything would crumble into confusion and conjecture.