BEIJING — Activity at China’s factories shrank at the fastest rate in at least three years in August, as domestic and export orders tumbled, increasing investors’ fears that the economy may be lurching toward a hard landing.

Even more worrying, China’s service sector, which has been one of the bright spots in the sputtering economy, also showed signs of cooling, a similar business survey said.

Hurt by soft demand, overcapacity and falling investment, the economy has also been buffeted by plunging shares and a shock devaluation in China’s currency, the renminbi, in a confluence of factors that is rattling global markets and could strain relations with China’s major trading partners.

The purchasing managers’ index for manufacturers released on Tuesday by China’s National Bureau of Statistics fell to 49.7 in August from 50 in July. That was the lowest level since August 2012, although expected by analysts in a Reuters poll. Readings above 50 points indicate growth; reading below indicate contraction.