Chapter 3: Chapter 3

Problem 65

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.

Short Answer

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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.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

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.

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Chapter 3: Chapter 3

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.

Short Answer

Step by step solution

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Step 1: 1. Existing limit

Step 2: 2. Continuous function at every point x

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More chapters from the book ‘Advanced Calculus’

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Chapter 2

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

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