Epsilon Delta Limit Definition

[MathSquareRoo]

Homework Statement

Prove lim x--> -1

1/(sqrt((x^2)+1)

using epsilon, delta definition of a limit

Homework Equations

The Attempt at a Solution

I know that the limit =(sqrt(2))/2

And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if 0<lx+1l<delta, then l[1/(sqrt((x^2)+1)]-(sqrt(2))/2l < epsilon. So we need to pick delta= ?

I'm not sure how to arrive at the delta. I know I have to work out what's inside the absolute values, but I'm getting stuck.

 

[SammyS]

Homework Statement

Prove lim x--> -1

1/(sqrt((x^2)+1)

using epsilon, delta definition of a limit

Homework Equations

The Attempt at a Solution

I know that the limit =(sqrt(2))/2

And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if 0<lx+1l<delta, then l[1/(sqrt((x^2)+1)]-(sqrt(2))/2l < epsilon. So we need to pick delta= ?

I'm not sure how to arrive at the delta. I know I have to work out what's inside the absolute values, but I'm getting stuck.

What have you tried?

Where are you stuck?

 

[MathSquareRoo]

I don't know how to simplify it at all. My thought was to maybe to get common denominator or and then multiply by the conjugate, but I don't know if this is correct. I got l[2-(2)^(1/2) *((x^2)+1)^(1/2)]/(2((x^2)+1)^(1/2)l. I'm sorry, it's hard to type it here, does any of that make sense?

 

[SammyS]

I don't know how to simplify it at all. My thought was to maybe to get common denominator or and then multiply by the conjugate, but I don't know if this is correct. I got l[2-(2)^(1/2) *((x^2)+1)^(1/2)]/(2((x^2)+1)^(1/2)l. I'm sorry, it's hard to type it here, does any of that make sense?

Yes, that makes some sense.

What you have is: \displaystyle \left|\frac{2-\sqrt{2}\sqrt{1+x^2}}{2\,\sqrt{1+x^2}}\right| which is equivalent to: \displaystyle \left|\frac{1}{\sqrt{1+x^2}}-\frac{\sqrt{2}}{2}\right|. That's a start.

Rationalize the numerator. One factor of the result will be (x + 1) .

Restrict δ to put a bound on the rest of the expression.

 

[MathSquareRoo]

When I try to rationalize, I get l(-2x^2)/[4((x^2)+1)^(1/2)+2(2)^(1/2)*((x^2)+1)]l

What did I do wrong?

 

[SammyS]

When I try to rationalize, I get l(-2x^2)/[4((x^2)+1)^(1/2)+2(2)^(1/2)*((x^2)+1)]l

What did I do wrong?

The numerator should be 2(1-x²) which is 1(1-x)(1+x).

The denominator is a mess.