Question

$$\text { Exponential limit Prove that } \lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}=e^{a}, \text { for } a \neq 0$$

Short answer

Question: Prove that $$\lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^x = e^a$$ for $$a \neq 0$$ using the definition of the exponential function. Answer: Using the definition of the exponential function, we demonstrated that replacing $$x$$ with $$a$$ in the definition results in the desired expression: $$\lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^x = e^a$$ for $$a \neq 0$$.

Step-by-step solution

Write down the given limit

We start by writing down the given limit: $$\lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^x$$.

Recognize the definition of the exponential function

Recall the definition of the exponential function: $$e^x = \lim_{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^n$$. We want to prove that the limit from Step 1 is equal to $$e^a$$, so let's compare these expressions.

Replace x with a in the exponential function definition

Replace $$x$$ with $$a$$ in the definition of the exponential function: $$e^a = \lim_{n \rightarrow \infty}\left(1+\frac{a}{n}\right)^n$$.

Compare the two expressions

Now we compare the limit in Step 1 with the expression for $$e^a$$ obtained in Step 3. The only difference is that, in Step 1, we have $$x$$ instead of $$n$$ in the limit expression. Since the variable is arbitrary, we can replace $$x$$ with $$n$$ in the limit expression from Step 1 to get: $$\lim_{n \rightarrow \infty}\left(1+\frac{a}{n}\right)^n$$

Conclude

Since both expressions from Step 4 represent the same limit, we can conclude that the given limit from Step 1 is equal to $$e^a$$: $$\lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^x = e^a$$ for $$a \neq 0$$.

Key concepts

Limit of a Function

Understanding the concept of a limit is crucial in calculus; it describes the value that a function approaches as the input (or index) approaches some value. In our exponential limit problem, we are interested in what happens as the variable, which in this case is denoted as \( x \), tends to infinity.

A limit can be intuitive initially; for example, if you keep halving the distance to a wall, the limit of your position is the wall itself, even if technically you might never touch it. Similarly, in our exercise, we explore what value the expression \( \left(1+\frac{a}{x}\right)^{x} \) approaches as \( x \) becomes very large.

The beauty of limits, especially in calculus, lies in their ability to make sense of values at points that are otherwise hard to define directly, such as infinity. This notion forms the bedrock upon which continuous mathematics is built and is an essential tool for solving many problems in mathematics and physics.

Exponential Function

An exponential function is denoted as \( e^x \), where \( e \) is the base of the natural logarithm and is approximately equal to 2.71828. It's a mathematical constant and an irrational number, meaning it cannot be expressed as a simple fraction, and its decimal representation goes on forever without repeating.

Exponential functions have the remarkable property that the rate of change (or derivative) of the function at any point is proportional to the value of the function at that point. In other words, they model growth or decay processes that are constant in percentage, not in absolute terms, which is common in natural phenomena like populations or interest calculations.

The notation \( e^x \) is significant because it represents a function that is its own derivative, a fundamental concept underlying many physical processes and one of the reasons why the exponential function is so pervasive in mathematics.

Definition of e

The number \( e \) is often referred to as Euler's number, after the Swiss mathematician Leonhard Euler. It's defined as the limit of \( \left(1+\frac{1}{n}\right)^n \) as \( n \) approaches infinity. This might seem abstract at first, but it encapsulates the idea of compounded growth happening an infinite number of times in a finite period.

This definition can also be extended to not just \( \left(1+\frac{1}{n}\right)^n \), but to more general forms, which is exactly what our exponential limit exercise is doing. By proving that \( \lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x} = e^{a} \), we're showing how this constant appears in various calculations of compound interest, population growth, and even in more abstract mathematical constructs.

Proofs in Calculus

Proofs are the foundation of mathematics; they provide the rigorous verification that certain mathematical statements are true. In calculus, proofs often involve limits, as they allow us to define derivatives and integrals.

The step-by-step solution provided in the exercise is an example of a direct proof, where we show that two expressions are equivalent by following a series of logical steps. We start by understanding the structure of the statement and then manipulating the expression to reveal its hidden truths.

For many students, proofs can be challenging because they require not only understanding the underlying concepts but also the ability to apply logical reasoning to arrive at a conclusion that is not immediately obvious. However, mastering the art of writing proofs is an essential skill for anyone looking to delve deep into the world of advanced mathematics.