Limit of a certain function of n as n goes to infinity

In summary, the limit in question is 3/e and can be solved by rewriting it as the limit of (1-1/n)^n and using the fundamental limit. Another approach is to rewrite it as 3/(1+1/n)^n and recognize it as a fundamental limit.
  • #1
ShizukaSm
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Homework Statement



[itex]\lim_{n->\infty} 3(\frac{n}{n+1})^n[/itex]

The Attempt at a Solution



Ok, I know that the answer is 3/e, because this limit was solved a year ago when I took calculus 1 by my teacher, and I foolishly copied only the answer, thinking I would never forget and have to go back.

I can't for the life of me understand how he did that, can someone help me, please?

Thanks in advance!
 
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  • #2
$$\frac{n}{n+1} = 1-\frac{1}{n+1}$$
Shifting the index by 1 (and ignoring the 3), the limit becomes $$\lim_{n \to \infty} \left(1-\frac{1}{n}\right)^n \left(1-\frac{1}{n}\right)^{-1}$$ which is easy to study.
 
  • #3
Another way to look at it:
$$3 \left(\frac{n}{n+1}\right)^n = \frac{3}{\left(\frac{n+1}{n}\right)^n} = \frac{3}{\left(1 + \frac{1}{n}\right)^n}$$
The expression in the denominator on the right hand side should look familiar.
 
  • #4
Oh yeah, of course! It's a fundamental limit, Thanks to you both, but I have to say that jbunniii's representation made it perfectly clear.
 

FAQ: Limit of a certain function of n as n goes to infinity

1. What is the meaning of "limit of a certain function of n as n goes to infinity"?

The limit of a certain function of n as n goes to infinity is the value that the function approaches as the input (n) approaches infinity. In other words, it is the value that the function gets closer and closer to, but may never reach, as the input increases without bound.

2. How is the limit of a function at infinity calculated?

The limit of a function at infinity is calculated by evaluating the function at larger and larger values of the input (n). If the function approaches a specific value as n goes to infinity, that value is the limit. If the function's values continue to increase or decrease without bound, the limit does not exist.

3. Can a function have multiple limits as n goes to infinity?

No, a function can only have one limit as n goes to infinity. If the function's values approach different values as n increases, then the limit does not exist.

4. What is the difference between a finite limit and an infinite limit?

A finite limit is a value that the function approaches as the input (n) increases without bound. An infinite limit, on the other hand, is a limit that does not exist because the function's values increase or decrease without bound as n goes to infinity.

5. How does the concept of a limit at infinity relate to real-world applications?

The concept of a limit at infinity is important in fields such as physics, engineering, and economics. It allows us to understand how a system or process behaves as time or other variables increase without bound. For example, in economics, the concept of a limit at infinity is used to analyze the long-term behavior of markets and economies.

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