Limit of (1-2/n)^n as n --> Infinity

In summary, we are trying to find the limit as n approaches infinity of (1-2/n)^n. We know that (1+1/x)^x approaches e as x approaches infinity. By letting n = -2x, we can manipulate the limit to use the known limit. We also use the fact that (1+1/x)^x approaches e as x approaches -infinity. Therefore, the limit of our original expression is e^-2, or 1/e^2.
  • #1
wimma
38
0

Homework Statement


Find the limit as n--> infinity of (1-2/n)^n


Homework Equations



We know (1+1/x)^x --> e as n--> infinity

The Attempt at a Solution



I worked it out as e^(-2) using log but I can't get it out using the fundamental limit above. I know it's the square of (1-1/x)^x (where we let x=n/2), just I don't know how to show that (1-1/x)^x --> 1/e. If you could let x |--> -x somehow I'd get the desired result using the limit laws but I'm not sure that's allowed.
 
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  • #2
Let n = -2x. This makes your limit
[tex]\lim_{-2x \to \infty} (1 + \frac{1}{x})^{-2x}[/tex]

With a bit of adjustment you can use the limit you know.
 
  • #3
but won't the parameter go to -infinity so we can't equate (1+1/x)^x to e?
 
  • #4
As it turns out,
[tex]\lim_{x \to -\infty} (1 + \frac{1}{x})^x~=~e[/tex]

Can you use this fact?
 

FAQ: Limit of (1-2/n)^n as n --> Infinity

What is the limit of (1-2/n)^n as n approaches infinity?

The limit of (1-2/n)^n as n approaches infinity is equal to 1/e. This means that as n gets larger and larger, the expression (1-2/n)^n approaches the value of 1 divided by the mathematical constant e.

How do you calculate the limit of (1-2/n)^n as n approaches infinity?

To calculate the limit of (1-2/n)^n as n approaches infinity, you can use the limit definition of the mathematical constant e, which is lim(1+1/n)^n as n approaches infinity. By substituting 2/n for 1/n, we get the expression (1-2/n)^n, which is equivalent to (1+1/n)^n. Therefore, the limit of (1-2/n)^n as n approaches infinity is equal to 1/e.

Does the limit of (1-2/n)^n as n approaches infinity exist?

Yes, the limit of (1-2/n)^n as n approaches infinity does exist. As n gets larger and larger, the expression (1-2/n)^n gets closer and closer to the value of 1/e. Therefore, the limit exists and is equal to 1/e.

What is the significance of the limit of (1-2/n)^n as n approaches infinity?

The limit of (1-2/n)^n as n approaches infinity has several important applications in mathematics and science. It is used in the calculation of compound interest, population growth, and radioactive decay. It also plays a key role in the definition and properties of the mathematical constant e.

Is there a visual representation of the limit of (1-2/n)^n as n approaches infinity?

Yes, there are several visual representations of the limit of (1-2/n)^n as n approaches infinity. One of the most common is a graph of the function y=(1-2/n)^n, where the x-axis represents n and the y-axis represents the value of the expression. As n approaches infinity, the graph approaches the horizontal line y=1/e. This visually illustrates the concept of approaching a limit as n gets larger and larger.

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