Limit of function with square roots

In summary, the conversation discusses the limit of \sqrt{n+ \sqrt{n}}-\sqrt{n} as n approaches infinity. By dividing each term by \sqrt{n}, the limit is found to be zero. The final answer is \lim_{n \to \infty} \sqrt{1 + \frac{1}{\sqrt{n}}} - 1. The conversation also questions why there are many problems involving limits, derivatives, and integrals in the precalculus section and clarifies the specific problem being asked.
  • #1
teng125
416
0
for sqr root of (n + sqr root (n) ) - sqr root (n),is the answer = zero or infinity so converges or diverges??
 
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  • #2
What would be the argument as to why it should be zero?
 
  • #3
if i divide each constant n by n itself then limit n to infinity and i'll get 1 - 1 =0 right??
 
  • #4
You mean, you'll have

[tex]n\left(\sqrt{\frac1n+\sqrt{\frac1{n^3}}}-\sqrt{\frac1n}}\right)[/tex]

and since 1/n3 ---> 0 much faster than 1/n, you'll end up with n[sqrt(1/n) - sqrt(1/n)]?
 
Last edited:
  • #5
no,just sqr root of 1/n without the power of 3
 
  • #6
if i divide every thing by n,i'll get sqr root [1 + (sqr root 1/n)] - sqr root (1)
what is the final answer??
 
  • #7
If you divide each term of
[tex]\sqrt{n+ \sqrt{n}}-\sqrt{n}[/tex]
by [itex]\sqrt{n}[/itex] you get
[tex]\sqrt{1+ \frac{1}{\sqrt{n}}}- 1[/tex]
What is the limit of that as n goes to infinity?


By the way, why are you posting so many problems involving limits, derivatives, and integrals in the precalculus section?
 
  • #8
is the answer = to zero??
 
  • #9
I have a question. It's not clear to tell that what you are asking for. Are you asking for this:
[tex]\lim_{n \rightarrow \infty} \sqrt{n - \sqrt{n}} - \sqrt{n}[/tex]
or this:
[tex]\lim_{n \rightarrow \infty} \frac{\sqrt{n - \sqrt{n}} - \sqrt{n}}{\sqrt{n}}[/tex]?
If it's the latter, then you are correct!
 

What is the definition of a limit of a function with square roots?

The limit of a function with square roots is the value that the function approaches as the input variable approaches a certain value. It represents the behavior of the function near a specific point on its graph.

How is the limit of a function with square roots calculated?

The limit of a function with square roots is calculated by evaluating the function at values that are closer and closer to the specified point, and observing the trend of these values. If the values approach a specific number, that number is the limit of the function.

Can the limit of a function with square roots have multiple values?

No, the limit of a function with square roots can only have one value. This value represents the behavior of the function near the specified point, and it cannot have multiple values.

Under what conditions does a limit of a function with square roots not exist?

A limit of a function with square roots does not exist if the values of the function at points close to the specified point do not approach a specific number, and instead have different values or do not approach any value at all.

Why is it important to understand limits of functions with square roots in scientific research?

Limits of functions with square roots are important in scientific research because they provide insight into the behavior of a function near a specific point. This can help in understanding the behavior of natural phenomena and making accurate predictions based on mathematical models.

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