Yes, you are correct.

Negating such statements is pretty easy. They usually have the form ∀x (P(x) → Q(x)) or ∃x (P(x) ∧ Q(x)) where Q(x) either is an atomic formula (equality, inequality, etc.) or in turn has the same form. We have the following equivalences.

¬(A → B) ⇔ A ∧ ¬B

¬(A ∧ B) ⇔ A → ¬B

¬∀x A(x) ⇔ ∃x ¬A(x)

¬∃x A(x) ⇔ ∀x ¬A(x)

Therefore,

¬(∀x (P(x) → Q(x))) ⇔ ∃x (P(x) ∧ ¬Q(x))

¬(∃x (P(x) ∧ Q(x))) ⇔ ∀x (P(x) → ¬Q(x))

Thus, to negate a proposition that starts with a sequence of quantifiers, you switch the quantifiers without changing the restrictions on quantified variables (P(x) above, e.g., x ∈ D, δ > 0 or |x - t| < δ) and negate only the last atomic formula.