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Chapter 4: Problem 16

As \(x \rightarrow \infty,\) the value of \(\left(1+\frac{1}{x}\right)^{x}\) approaches _____.

Short Answer

Expert verified
The value approaches \( e \).

Step by step solution

01

Understanding the Expression

The given expression is \(\left(1+\frac{1}{x}\right)^{x}\). To find the limit as \( x \rightarrow \infty\), consider the behavior of the expression as \( x \) becomes very large.
02

Definition of Euler's Number

Recall that the limit \( \lim_{x\to\infty} \left(1+\frac{1}{x}\right)^{x} \) is a known limit and equals the constant \( e \), which is approximately 2.71828.
03

Applying the Limit

Using the definition, we have: \(\lim_{x\to\infty} \left(1+\frac{1}{x}\right)^{x} \) = \( e \). This shows that as \( x \rightarrow \infty\), \( \left(1+\frac{1}{x}\right)^{x} \) approaches \( e \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euler's number
Euler's number, denoted as \(e\), is a fundamental mathematical constant. It is approximately 2.71828, but its exact value continues without repeating or terminating. Euler's number plays a crucial role in calculus, particularly in the study of growth and decay processes. It is also the base of the natural logarithm.
One of the most important properties of \(e\) is its relationship with the expression \( \left(1 + \frac{1}{x}\right)^x \). As \(x\) approaches infinity, this expression approaches the value \(e\). This property is essential for understanding the behavior of exponential functions and their limits. The formula can be written as:
\[ e = \lim_{x\to\infty} \left(1+\frac{1}{x}\right)^{x}\] This limit is fundamental because it provides a way to define \(e\) using limits, which is a key concept in calculus.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

Graph the following functions on the window [-3,3,1] by [-1,8,1] and comment on the behavior of the graphs near $$ \begin{array}{l} x=0 \\ \mathrm{Y}_{1}=e^{x} \\ \mathrm{Y}_{2}=1+x+\frac{x^{2}}{2} \\ \mathrm{Y}_{3}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6} \end{array} $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}=-7 e^{x}\)

Given \(f(x)=b^{x},\) then \(f^{-1}(x)=\) _____ for \(b>0\) and \(b \neq 1\).

Solve for the indicated variable. \(Q=Q_{0} e^{-k t}\) for \(k\) (used in chemistry)
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