Proving Continuity at a Point using (ε,δ) Method

[ferret93]

I'm working on a problem as part of exam revision, but I've run into a bit of trouble so far. The problem is;
Give an (ε,δ) proof that f(x) = 1/\sqrt{10 - x^2} is continuous at x = -1

The attempt at a solution
So far what I've gotten is f(x) - f(-1) = 1/(\sqrt{10 - x^2}) - 1/3
= (3 - (\sqrt{10 - x^2}))/(3\sqrt{10 - x^2})
= ((x^2) - 1)/(3\sqrt{10 - x^2})

Then from here I've gotten -2 < x < 0 → 10 - x^2 > 6
i'm lost from where to go from here though i just can't see a way though, any help will be greatly appreciated.

 

[tiny-tim]

welcome to pf!

hi ferret93! welcome to pf!

= ((x^2) - 1)/(3\sqrt{10 - x^2})

the standard trick is to write that as (x+1) times the rest …

you use ε,δ to minimise (x+1), and some other limit to kepp the rest bounded