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Proving limit using epsilon delta definition, three options

Proving $\lim \limits_{x\to \infty} (a^2 – b^2)^{1/2} = \sqrt{a^2-b^2}$ using the following three ways:

$\epsilon$-delta definition of limit.
Limit comparison.
L’Hopital’s Rule.

Here’s what I did:
$|(a^2 – b^2)^{1/2} – \sqrt{a^2 – b^2}| = |(a^2 – b^2)^{1/2} – a^2 + b^2|| = |(a^2 – b^2)^{1/2} – a^2| + |b^2| = |(a – b)(a + b)|^{1/2} + |b|$
The second equality comes from the fact that $|a – b| = |a + b|$ and the absolute value function.
However, I’m not really sure whether this is the correct way of doing it. I don’t think this problem is as easy as the book has it, since there’s the absolute value function.
Which of
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