zögern
to hesitate
To pause or wait for a moment before doing or saying something, often out of uncertainty, doubt, or indecision.
Er zögerte kurz, bevor er die Tür öffnete.
He hesitated briefly before opening the door.

to hesitate
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to hesitate
To pause or wait for a moment before doing or saying something, often out of uncertainty, doubt, or indecision.
Er zögerte kurz, bevor er die Tür öffnete.
He hesitated briefly before opening the door.

'Zögern' (to hesitate) is often used in two patterns. You can say 'zögern, etwas zu tun' (to hesitate to do something; with an infinitive and 'zu'). You can also say 'mit etwas zögern' (to delay something; with the preposition 'mit' and dative).
Sie zögerte, die Wahrheit zu sagen.
She hesitated to tell the truth.
The expression 'ohne zu zögern' (without hesitating) is very common and means 'immediately' or 'without delay.' It expresses that someone carries out an action without doubt or pause.
Er half der alten Dame ohne zu zögern über die Straße.
He helped the old lady across the street without hesitating.
The negative request 'Zögern Sie nicht, ...' (Do not hesitate to ...) is a very polite and common way to encourage someone to do something, for example to get in touch.
Zögern Sie nicht, bei Fragen anzurufen.
Do not hesitate to call if you have any questions.
to ask / request
'sich bitten lassen' and 'zögern' both have to do with waiting before doing something. 'zögern' is more general: you do not do something immediately because you are unsure or still thinking. 'sich bitten lassen' is used in a special situation: someone is asked or invited to do something, but does not join in right away and waits until others ask again. This can sound playful, polite, or also somewhat critical. So 'zögern' describes the waiting itself, while 'sich bitten lassen' describes hesitating after a request or invitation.

Nach der Nachricht zögert Maria noch mit ihrer Antwort.
After the message, Maria is still hesitating over her reply.
Nach der Einladung ließ sich Paul nicht lange bitten und kam mit.
After the invitation, Paul did not need much persuading and came along.