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Magazine Chapter 3: Problem 98
$$\text { Show that } \lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x} \text { for any } x>0$$.
Short Answer
Expert verified
The limit \( \lim_{{n \to \infty}} \left(1+\frac{x}{n}\right)^n = e^x \) for any \( x > 0 \).
Step by step solution
01
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 \( x > 0 \).
02
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.
03
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.
04
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:\[\ln \left( 1 + \frac{x}{n} \right) \approx \frac{x}{n}\]
05
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 \( \lim_{{n \to \infty}} \ln \left(1 + \frac{x}{n} \right)^n = x \).
06
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 \( e^x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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 \(x\).
