# Limits using basic analysis theorems and logic?

• broegger
In summary, Joe found an example of a clever way of dealing with limits involving differences like these in the back of the book. He suggests using the binomial theorem to simplify the expression.
broegger
Hi,

I need help again. How can I show that

$$\sqrt{n^2+2n}-n \rightarrow 1$$​

for $$n\rightarrow\infty$$ using basic analysis theorems and logic? Any ideas?

I bet if you checked your book, you'd find an example of a clever way of dealing with limits involving differences like these...

try doing something to it that involves its conjugate...

Rewrite $$\sqrt{n^2+2n}-n$$ as

$$[\sqrt{(n+1)^2 - 1^2} -(n+1)] + 1$$

Thus you need to show that

$$\sqrt{(n+1)^2 - 1^2} -(n+1) \rightarrow 0$$

as n to infinity. But if you look at the way it is written, you will see that it looks suspiciously like the difference between the length of two sides of a right triangle as one of the points on the triangle goes to infinity.

Carl

Thanks guys. Hurkyl was right; there actually was an example dealing with this in the back of the book. I can't believe I missed it.

What is the solution?

We want to show that $$\sqrt{n^2+2n} - (n+1) \rightarrow 0$$, so we rewrite the expression:

$$\sqrt{n^2+2n} - (n+1) = \frac{\left(\sqrt{n^2+2n} - (n+1)\right)\left(\sqrt{n^2+2n} + (n+1)\right)}{\sqrt{n^2+2n} + (n+1)} = \frac{-1}{\sqrt{n^2+2n} + (n+1)}$$

It is obvious that the final expression tends to 0 as $$n\rightarrow\infty$$.

I solved it thus:

(n^2 + 2n)^1/2 = (n^2)^1/2 * (1+2/n)^1/2

therefore this = n(1+2/n)^1/2

now expand the bracket using taylor's theorem to get

(1+2/n)^1/2= 1+(1/n)-(1/n^2)+...

therefore multiplying by n in the above you have n+1-(1/n) as n tends to infinity the (1/n) becomes irrelevant and we are left with n+1. Just subtract the n from the original expression and you obtain the desired result, 1.

Take care, Joe

Gosh, that looks complicated!

I would have thought using gDogg's suggestion and writing
$$\sqrt{n^2+2n}-n= \frac{(\sqrt{n^2+2n}-n)(\sqrt{n^2+2n}+n)}{\sqrt{n^2+2n}+n}$$
$$= \frac{2n}{\sqrt{n^2+ 2n}+n}$$

would be much simpler.

Yes... he could have used the binomial theorem!

$$(n^2 + 2n)^{1/2} = (n^2)^{1/2} + \frac{1/2}{1} (n^2)^{-1/2} (2n) + \frac{(1/2) * (-1/2)}{1 * 2} (n^2)^{-3/2} (2n)^2 + \cdots = n + 1 + O\left(\frac{1}{n}\right)$$

(since n² > 2n when n grows large)
(Hrm, did he make a mistake in the third term, or did I?)

Last edited:

## What are limits in mathematics?

Limits in mathematics refer to the value that a function approaches as the input approaches a certain value. It represents the behavior of a function near a specific point.

## What is the limit notation?

The limit notation is written as lim(x→a) f(x), where "lim" stands for limit, "x→a" means that the input is approaching a specific value "a", and "f(x)" is the function being evaluated.

## What are the basic analysis theorems related to limits?

The basic analysis theorems related to limits are the Squeeze Theorem, the Intermediate Value Theorem, and the Convergence-Divergence Theorem. These theorems provide useful tools for evaluating limits and determining the behavior of a function.

## How is logic used in determining limits?

In determining limits, logic is used to analyze the behavior of a function and make inferences based on the given information. Through logical reasoning, it is possible to determine the value of a limit and prove its existence or non-existence.

## What is the difference between a one-sided and two-sided limit?

A one-sided limit only considers the behavior of a function from one direction, either from the left or the right of a specific point. A two-sided limit, on the other hand, considers the behavior of a function from both directions and requires that the function approaches the same value from both sides.

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