Why is $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e$?

  • Thread starter Zarlucicil
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In summary, the conversation discusses a limit question about the limit as x approaches infinity of (1+1/x)^x, which equals e according to Wolframalpha. The reasoning for this answer involves using the limit definition of e and taking the derivative of ln(1+t). The individual asking the question wonders why the same reasoning does not apply when using log base 10 instead of ln. However, they realize their mistake in taking the derivative of log(1+t) and apologize for any inconvenience.
  • #1
Zarlucicil
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This is a pretty basic limit question regarding the limit,

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e [/tex]

Wolframalpha gives the following reasoning for this answer:

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e^{\lim_{x \rightarrow \infty} x\ln{(1+\frac{1}{x})}} = e^{\lim_{t \rightarrow 0} \frac{\ln{(1+t)}}{t}} = e^{\lim_{t \rightarrow 0} \frac{ \frac{d\ln{(1+t)}}{dt}}{\frac{d}{dt}t}}= e^{\lim_{t \rightarrow 0} \frac{1}{1+t}} = e^1 [/tex]

My question is, by the same reasoning, why is the following not true? (where log is log base 10)

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = 10^{\lim_{x \rightarrow \infty} x\log{(1+\frac{1}{x})}} = 10^{\lim_{t \rightarrow 0} \frac{\log{(1+t)}}{t}} = 10^{\lim_{t \rightarrow 0} \frac{ \frac{d\log{(1+t)}}{dt}}{\frac{d}{dt}t}}= 10^{\lim_{t \rightarrow 0} \frac{1}{1+t}} = 10^1 [/tex]

Am I missing something??
 
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  • #2
Wooooow, hold on, I understand what I did wrong...

The derivative of [tex] \log{(1+t)} [/tex] is NOT [tex] \frac{1}{1+t} [/tex] but rather [tex] \frac{1}{(1+t)\ln{10}} [/tex]

Sorry for wasting the time of those who've read this.
 

Related to Why is $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e$?

1. What is the significance of the limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$?

The limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ has significant implications in calculus and real analysis. It is used to define the number e, which is a fundamental constant in mathematics. This limit also plays a crucial role in the study of exponential growth and decay.

2. How is the limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ related to the number e?

The limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ is the definition of the number e. As the value of x approaches infinity, the expression $(1+\frac{1}{x})^x$ approaches the value of e. In other words, e is the number that the expression approaches as x becomes infinitely large.

3. What is the intuition behind the limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ approaching e?

The limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ can be thought of as a continuous compounding interest rate. As the number of compounding periods (x) becomes infinitely large, the resulting value approaches the constant e. This means that e represents the maximum amount that can be earned from continuously compounding interest.

4. Can the limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ be evaluated using a calculator?

No, most calculators have a limit to the number of decimal places they can display, so they cannot accurately calculate the value of the limit as x approaches infinity. However, some advanced calculators or computer software can approximate the value of e by using larger and larger values of x.

5. What are some real-world applications of the limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$?

The limit $\lim_{x \rightarrow \infty} (1+\frac{1}{x})^x$ has many real-world applications, such as in finance, physics, and biology. In finance, it is used to calculate the future value of an investment with continuously compounding interest. In physics, it is used to model exponential growth and decay, such as in radioactive decay. In biology, it is used to model population growth and the spread of diseases.

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