# Proving Limit of r/n as n Approaches Infinity

• gtfitzpatrick
In summary, the limit of r/n as n approaches infinity is 0, which means that as n gets larger, the value of r/n gets closer to 0. To prove this limit, the formal definition is used, which involves showing that for any given positive number ε, there exists a corresponding value of n where the absolute value of r/n is less than ε. L'Hôpital's rule cannot be used in this proof, as it only applies to indeterminate forms. The limit of r/n as n approaches infinity is always 0, as the denominator (n) grows much faster than the numerator (r). This proof is significant in understanding the behavior of a function or sequence as it approaches infinity, and has various
gtfitzpatrick

## Homework Statement

prove that lim(n$$\rightarrow\infty$$)(r1/n) = 1 for r> 0

## The Attempt at a Solution

let $$\epsilon$$ > 0 be given we need to find n0 $$\in$$ N such that

$$\left|$$r1/n - 1 $$\left|$$ < $$\epsilon$$

but not really sure where to go from here?

What if the problem was a little simpler -- proving that the limit as n approaches 0 of r^n equals 1. How would you go about doing that?

1 < L $$\leq$$ r1/n

implies

1$$\leq$$ Ln $$\leq$$ r

i'm not sure how this follows?

## 1. What is the limit of r/n as n approaches infinity?

The limit of r/n as n approaches infinity is 0. This means that as n gets larger and larger, the value of r/n gets closer and closer to 0.

## 2. How do you prove the limit of r/n as n approaches infinity?

To prove the limit of r/n as n approaches infinity, we use the formal definition of a limit. This involves showing that for any given positive number ε, there exists a corresponding value of n such that when n is greater than this value, the absolute value of r/n is less than ε.

## 3. Can you use L'Hôpital's rule to prove the limit of r/n as n approaches infinity?

No, L'Hôpital's rule cannot be used to prove the limit of r/n as n approaches infinity. This rule is only applicable when dealing with limits of indeterminate forms, which do not apply in this case.

## 4. Is the limit of r/n as n approaches infinity always 0?

Yes, the limit of r/n as n approaches infinity is always 0. This is because as n gets larger and larger, the denominator (n) grows much faster than the numerator (r), causing the overall fraction to approach 0.

## 5. What is the significance of proving the limit of r/n as n approaches infinity?

Proving the limit of r/n as n approaches infinity is important in understanding the behavior of a function or sequence as it approaches infinity. It helps us determine the ultimate behavior of the function and can be used in various mathematical and scientific applications.

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