at the point of meaning

On the verge of

[American slang]
prep. Very near to; almost at or in. When Mary broke her favorite bracelet, she was at the point of tears. The boy hurt in the accident lay at the point of death for a week, then he got well.
Compare: ABOUT TO¹, ON THE POINT OF.

Examples

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1. alongshore bars are produced at the point of wave break.
2. the root hair cell wall invaginates at the point of contact.
3. the thickness of this film is very small at the point of contact.
4. his temperament might be said to be just at the point of maturity.
5. a traverse load on a beam causes a deflection or change in elevation at the point of application.