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Acercarse vs. aproximarse

The Spanish verbs acercarse and aproximarse both mean to approach or to draw near to. However, their usage can often depend on the context.

Acercarse

A2
Acercarse generally means to move closer or to get near to. This word often indicates some sort of physical movement towards something or someone.
Voy a acercarme a la ventana para ver mejor.
(I'm going to move closer to the window to see better.)
Él se acercó a y me dijo al oído que tenía un secreto.
(He drew near to me and whispered that he had a secret.)
¡No te acerques al fuego!
(Do not approach the fire!)

Aproximarse

B1
Aproximarse can also mean to approach or to draw near. However, this term is often used in more abstract senses, like time or similarity, rather than physical movement.
Se aproxima el fin de semana.
(The weekend is approaching.)
Ella se estaba aproximando a la verdad.
(She was getting close to the truth.)
El número de visitantes se aproxima a mil.
(The number of visitors approaches one thousand.)

Summary

Acercarse and aproximarse both refer to the act of moving closer or drawing near. The difference lies in their context of use. While acercarse typically denotes physical movement towards something or someone, aproximarse is often used in more abstract terms such as time, numbers, and concepts.