Epsilon-delta proof

[Ming1015]

Hello, I would ike to ask how to prove lim x→√3 (1/√3) = ⅓. I'm struggling to assume the value of δ in terms of ε when I found out that |x-√3| = ε(3x²/x+√3) with the formula |f(x) - L| < ε when |x-x₀| < δ, but I think that there's no way I could assume δ in terms of x too. May I know that is there any methods in solving it?

 

[pka]

Hello, I would ike to ask how to prove lim x→√3 (1/√3) = ⅓. I'm struggling to assume the value of δ in terms of ε when I found out that |x-√3| = ε(3x²/x+√3) with the formula |f(x) - L| < ε when |x-x₀| < δ, but I think that there's no way I could assume δ in terms of x too. May I know that is there any methods in solving it?

How can we possibly help. You never told us what the function is.
What is \(\bf f(x)=?\)

 

[Ming1015]

How can we possibly help. You never told us what the function is.
What is \(\bf f(x)=?\)

Hello, sir. I'm sorry, I wrote the limit wrongly. It should be lim x→√3 (1/x²) = ⅓ Just for confirmation, I'm using x₀ = √3, f(x) =1/x² and then L =⅓.

 

[pka]

Hello, sir. I'm sorry, I wrote the limit wrongly. It should be lim x→√3 (1/x²) = ⅓ Just for confirmation, I'm using x₀ = √3, f(x) =1/x² and then L =⅓.

Start with,
\(|x-\sqrt3|<1\\-1<x-\sqrt3<1\\\sqrt3-1<x<\sqrt3+1\)
From that start you must build up the expression \(\left|\dfrac{1}{x^2}-\dfrac{1}{3}\right|=\left|\dfrac{3-x^2}{3x^2}\right|<?\)
Then pick your \(\delta=\min\left\{\dfrac{\varepsilon}{?},1\}\right\}\) and proceed.