Why Is the Limit of This $\ln$ Sequence Incorrect?

In summary, the limit of a sequence is the value that the terms of the sequence approach as the index increases without bound. To find the limit, various techniques can be used such as algebraic manipulation, substitution, or rules of limits. A graphing calculator or online calculator can also be helpful. The limit of a ln sequence is the same as the limit of a general sequence, and it may be a finite number or may not exist. Some common properties of ln sequences include being always positive, approaching zero as the index increases without bound, and decreasing as the index increases. Ln sequences can have both finite and infinite limits, depending on the specific sequence.
  • #1
tmt1
234
0
If I have this sequence

$$a_n = \ln\left({\frac{n}{n^2 + 1}}\right)$$

I need to find:

$$ \lim_{{n}\to{\infty}} \ln\left({\frac{n}{n^2 + 1}}\right)$$

Shouldn't I be able to find the limit of$$ \lim_{{n}\to{\infty}} \frac{n}{n^2 + 1}$$ (which is $0$) and then substitute the result of that into the original limit and get the answer there?

So if I substitute in 0 I would get

$$ \lim_{{n}\to{\infty}} \ln\left({0}\right)$$

which would be negative $\infty$. However this is the incorrect answer.
 
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  • #2
Why do you think that the result is incorrect?
 

FAQ: Why Is the Limit of This $\ln$ Sequence Incorrect?

What is a limit of a sequence?

A limit of a sequence is the value that the terms of the sequence approach as the index of the terms increases without bound. In other words, it is the value that the terms of the sequence get closer and closer to, but never actually reach.

How do you find the limit of a sequence?

To find the limit of a sequence, you can use various techniques such as algebraic manipulation, substitution, or rules of limits. You can also use a graphing calculator or online calculator to help visualize and calculate the limit.

What is the limit of a ln sequence?

The limit of a ln sequence is the same as the limit of a general sequence. It is the value that the terms of the sequence approach as the index increases without bound. In the case of a ln sequence, the limit may be a finite number or may not exist.

What are some common properties of ln sequences?

Some common properties of ln sequences include the fact that they are always positive, they approach zero as the index increases without bound, and their terms decrease as the index increases.

Can ln sequences have infinite limits?

Yes, ln sequences can have infinite limits. For example, the sequence ln(n) approaches infinity as the index n increases without bound. However, it is also possible for ln sequences to have finite limits, such as the sequence ln(1/n) which approaches zero as n increases without bound.

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