France realized that the united states might win the war after the american victory at___

At Saratoga.

Explanation:

The Battle of Saratoga was fought in September-October  of 1777. It is actually two battles that were fought.

It followed a failed campaign by the Continental Army to invade Canada and control some territory. They were beaten by the British. But the two enemies met again in Saratoga.

The first battle did not produce a decisive result. But in the second engagement, the Continental Army soundly defeated the British, causing many casualties and losses.

Saratoga was a turning point in the early phase of the war. It persuaded the French to help Americans, they saw Americans had a chance to win the war.