(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove (using epsilon-delta definition only) that the limit of the following expression:

[itex]sqrt(n^2+n)-n[/itex]

is 1/2, as n tends towards infinity.

2. Relevant equations

For any ε > 0, there exists some natural number N, such that:

n > N gives|f(n) - L| < ε

3. The attempt at a solution

Multiply by the conjugate pair and simplify to obtain:

[itex]n/[sqrt(n^2+n)+n][/itex]

Taking the expression to be compared against ε

|[itex]n/[sqrt(n^2+n)+n] - 1/2[/itex]| and finding a common denominator gives:

[itex]|2n/2[sqrt(n^2+n)+n] - [sqrt(n^2+n)+n]/2[sqrt(n^2+n)+n]|[/itex]

[itex]|[n-sqrt(n^2+n)] / 2[sqrt(n^2+n)+n]|[/itex]

The numerator is always negative and the denominator is always positive, so the expression can be re-written as:

[itex][sqrt(n^2+n) - n][/itex] / [itex]2[sqrt(n^2+n)+n][/itex]

I chose to split this up into two terms:

[itex]sqrt(n^2+n)[/itex] / [itex]2[sqrt(n^2+n)+n][/itex] minus

[itex]n[/itex] / [itex]2[sqrt(n^2+n)+n][/itex]

I have an expression in the form a/b-c/d. Decreasing "b", decreasing "c" and increasing "d" both increase the value of the expression, thus the expression below is strictly greater than the

expression above.

[itex]sqrt(n^2+n)[/itex] / [itex]2[sqrt(n^2+n)][/itex] minus

[itex]1[/itex] / [itex]2[sqrt(n^2+n^2)+n][/itex]

This simplifies to [itex]1/2[/itex] - [itex]1[/itex] / [itex]n(2sqrt(2)+1)][/itex]

Both terms in this expression are positive. Thus, changing the subtraction to addition will strictly increase the value of the entire quantity.

Thus, comparing

[itex]1/2[/itex] + [itex]1[/itex] / [itex]n(2sqrt(2)+1)[/itex] against ε is enough.

Solving for n results in some mess that results in:

n > (positive constant) / (ε - 1/2)

Clearly, something has gone awry. Consideration of an arbitrarily small ε reveals the RHS to be negative. That would imply that the first term in my sequence is already arbitrarily close to my limit.

Where did I go wrong?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Epsilon-delta proof of limits

**Physics Forums | Science Articles, Homework Help, Discussion**