Problem 98 - University Calculus: Early Transcendentals

Question

$$\text { Show that } \lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x} \text { for any } x>0$$.

Step-by-Step Solution

Answer

The limit \( \lim_{{n \to \infty}} \left(1+\frac{x}{n}\right)^n = e^x \) for any \( x > 0 \).

Step 1: Introduce the Limit Expression

The given limit expression is \( \lim_{{n \to \infty}} \left( 1 + \frac{x}{n} \right)^n \). We need to show that this limit equals \( e^x \) for any \( x > 0 \).

Step 2: Use the Definition of the Exponential Function

Recall the definition of the exponential function: \( e^x = \lim_{{n \to \infty}} \left( 1 + \frac{x}{n} \right)^n \). We will show that this limit satisfies the properties of the exponential function through manipulation.

Step 3: Perform Logarithmic Transformation

Consider taking the natural logarithm of the limit expression:\[\ln \left(\lim_{{n \to \infty}} \left(1 + \frac{x}{n} \right)^n \right) = \lim_{{n \to \infty}} n \ln \left( 1 + \frac{x}{n} \right)\]This step simplifies the process by converting the expression into an exponential form.

Step 4: Approximate Using Taylor Expansion

Use the first-order Taylor series expansion for \( \ln(1+u) \approx u \) when \( u \) is very small.Here \( u = \frac{x}{n} \), therefore:\[\ln \left( 1 + \frac{x}{n} \right) \approx \frac{x}{n}\]

Step 5: Evaluate the Simplified Limit

Substitute the approximation into the limit expression:\[\lim_{{n \to \infty}} n \cdot \frac{x}{n} = \lim_{{n \to \infty}} x = x\]Hence, the limit becomes \( \lim_{{n \to \infty}} \ln \left(1 + \frac{x}{n} \right)^n = x \).

Step 6: Conclude with the Exponential Function

Since \( \ln(a) = b \) implies \( a = e^b \), we find:\[\ln \left( \lim_{{n \to \infty}} \left(1 + \frac{x}{n} \right)^n \right) = x \Rightarrow \lim_{{n \to \infty}} \left(1 + \frac{x}{n} \right)^n = e^x\]This confirms that the original limit indeed converges to \( e^x \).

Key Concepts

Exponential Functions

Exponential functions are a fundamental component of mathematics and occur frequently in natural phenomena. The standard exponential function is written as \(e^x\), where \(e\) is a mathematical constant approximately equal to 2.71828. This function is characterized by its unique properties:

• It is the only function that is its own derivative, meaning \( \frac{d}{dx}e^x = e^x \).


• For any real number \(x\), the function \(e^x\) represents exponential growth (if \(x > 0\)) or decay (if \(x < 0\)).


• It provides the continuous compound interest formula, which is important in finance and natural sciences.

When approaching problems involving limits and exponential functions, understanding their properties can drastically simplify calculations. Specifically, showing \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \) uses the core idea that as small increments repeatedly compound, they approach this natural exponential result.

Natural Logarithms

Natural logarithms are the inverse of exponential functions. Given a number \(y\), the natural logarithm, denoted \(\ln(y)\), is the power to which \(e\) must be raised to obtain \(y\). For example, \(e^3 = e^x\) means \(\ln(e^3) = 3\).

• Natural logarithms convert multiplicative relationships into additive ones, which is vital for simplifying exponential expressions.


• This property is often used to maximize or minimize functions, especially when exponential growth or decay is involved.


• In the original exercise, we use the property: \(\ln(a^b) = b \ln(a)\), which helps to switch problematic power functions into more manageable multiplication.

By using natural logarithms, the step-by-step solution was able to transform an exponential limit expression into a format that allows for easier manipulation and evaluation.

Taylor Series Expansion

The Taylor series expansion is a mathematical tool used to approximate functions by polynomials. For a function \(f(x)\) that is infinitely differentiable around a point \(a\), its Taylor series is expressed as:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots\]When \(x\) is very close to \(a\), higher-order terms become negligible, and the series provides an accurate approximation.

• The Taylor expansion of \(\ln(1+u)\) around \(u=0\) is \(\ln(1+u) \approx u\), for very small \(u\).


• This expansion helps simplify complex logarithmic expressions, particularly in limit problems.


• In exercises involving limits, using the Taylor series allows us to see how functions behave near points of interest and aids in revealing that \(\lim_{n \to \infty} n \ln\left(1 + \frac{x}{n}\right)\) simplifies to \(x\).

Hence, Taylor series provides the bridge to approximate and solve limits involving logarithms and exponential functions more effectively.