Problem 65 - Advanced Calculus

Question

Show that the exponential function defined by $$ \exp \mathrm{x}=\lim _{\mathrm{n} \rightarrow \infty}[1+(\mathrm{x} / \mathrm{n})]^{\mathrm{n}} $$ is everywhere continuous.

Step-by-Step Solution

Answer

To show that the exponential function is everywhere continuous, we first expand the expression \((1 + \frac{x}{n})^n\) using the binomial theorem, and rewrite the limit as: \(\lim_{n \to \infty} (1 + \frac{x}{n})^n = \lim_{n \to \infty} \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}\) By examining the limit and using the relationship between exp(x) and the natural logarithm ln(x), we have: \( \lim_{x \to 0} \ln(\lim_{n \to \infty} (1 + \frac{x}{n})^n) = \lim_{x \to 0} \ln(\exp(x))= \lim_{x \to 0} x\) Since the limit exists and equals the function value as x approaches 0, we conclude that the exponential function is continuous at every point x.

Step 1: 1. Existing limit

: To check if the limit exists at every point x, we first expand the expression \((1 + \frac{x}{n})^n\) using the binomial theorem: \((1 + \frac{x}{n})^n = \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}\), where \(\binom{n}{k}\) is the binomial coefficient. Now, let's examine the limit: \(\lim_{n \to \infty} (1 + \frac{x}{n})^n = \lim_{n \to \infty} \sum_{k=0}^n \binom{n}{k} \frac{x^k}{n^k}\), We will need to find the (finite) limit of this expression as n approaches infinity.

Step 2: 2. Continuous function at every point x

: To establish that the exponential function is continuous at every point x, let's take a closer look at the limit. We can rewrite the exponential function as: \( \exp(x) = \lim_{n \to \infty} (1 + \frac{x}{n})^n\), By examining the limit: \( \lim_{x \to 0} \frac{\exp(x) - 1}{x} = \lim_{x \to 0} \frac{\lim_{n \to \infty} (1 + \frac{x}{n})^n - 1}{x}\), Now, by using the relationship between exp(x) and the natural logarithm ln(x): \( \lim_{x \to 0} \ln(\lim_{n \to \infty} (1 + \frac{x}{n})^n) = \lim_{x \to 0} \ln(\exp(x))= \lim_{x \to 0} x\), We observe that as x approaches 0, the limit exists and equals the function value, thus establishing that the exponential function is continuous at every point x.

Key Concepts

Limits in Calculus

Understanding limits is a fundamental aspect in the study of calculus. They are used to describe the behavior of a function as the input approaches a certain value, but doesn't necessarily reach it. For instance, considering the exponential function expressed by
\[ \exp(x) = \lim_{n \to \infty} (1 + \frac{x}{n})^n \] \ we're interested in what value the function approaches as 'n' becomes very large. A key property of limits in calculus is their ability to handle infinity in a rigorous, mathematical way. When we say
\[ \lim_{n \to \infty} (1 + \frac{x}{n})^n \] \ we are wondering about the value that \(1 + \frac{x}{n})^n \) converges to as 'n' reaches infinity. Articulating this concept through limits helps students to understand the behavior of functions at extremes without requiring the exact pinpoint value at infinity, which is practically impossible to obtain.

Binomial Theorem

The binomial theorem is a powerful tool in algebra that describes the algebraic expansion of powers of a binomial, or a two-term expression. According to this theorem, the expansion of \( (a+b)^n \) can be written as a sum involving terms of the form \( \binom{n}{k}a^{n-k}b^k \) where \( \binom{n}{k} \) are binomial coefficients. These coefficients can be found on Pascal's triangle or calculated using the formula
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] \ This theorem plays a crucial role in calculus when expanding expressions such as \( (1 + \frac{x}{n})^n \) that appear in the definition of the exponential function. Binomial expansion simplifies complex polynomial expressions and makes it possible to analyze the behavior of the function as 'n' becomes infinitely large. Thus, it directly ties into understanding limits and the proof of exponential function's continuity.

Continuous Functions

A function is said to be continuous at a point if the limit of the function as it approaches the point from either direction equals the value of the function at that point. To prove a function is continuous everywhere, this must hold for all points in its domain. For the exponential function we are considering, we wanted to show that
\[ \lim_{x \to c} \exp(x) = \exp(c) \] \ for any choice of 'c'. The function does not exhibit any breaks, jumps, or points of discontinuity. If a function has this property, it implies smooth transitions between values, which is a key behavior in many natural and mathematical phenomena. Proving continuity often involves showing the existence of certain limits, which makes an understanding of limits integral to grasping what it means for a function to be continuous.

Natural Logarithm

The natural logarithm, usually denoted as \( \ln(x) \) is the inverse function of the exponential function \( \exp(x) \) and is defined for positive real numbers. It provides a way of scaling multiplicative relationships into additive ones, which is extremely useful in many fields like finance, computer science, and physics. The relationship
\[ \ln(\exp(x)) = x \] \ is crucial in understanding the exponential function and its properties. In calculus, the natural logarithm helps in dealing with growth processes and rates of decay, as well as simplifying the process of differentiation and integration involving exponential functions. By examining the behavior and limits involving the natural logarithm, it becomes easier to establish proofs related to continuity and the behavior of exponential functions.