OLYMPIA — Private-sector job growth has pushed Washington’s unemployment rate to the lowest point since February 2009, officials said Wednesday.The November jobless rate of 8.7 percent was down from 9.1 percent in October, according to the Employment Security Department. The state added some 12,100 jobs — more than any month since the official start of the recession at the end of 2007. Clark County’s October unemployment rate was likely around 11.4 percent; an official estimate will be released by state officials on Dec. 20. The county’s jobless rate was 11.6 percent in September. Clark County jobless figures lag statewide unemployment numbers by about a month, because of the extra time involved in incorporating data reported by Oregon employers of Southwest Washington residents.“We are upbeat. This is good news, but we are being cautious at the same time,” said Greg Morgan, a labor market economist at the Employment Security Department. “Just one month isn’t enough to feel comfortable saying we’re out of the woods just yet.”The state has been adding jobs regularly over the past year, but it has usually come in smaller chunks. Morgan said it was too early to say whether the growth was the start of a trend, and noted that the November numbers are preliminary and may be revised. The October jobless number was revised upward from 9 percent to 9.1 percent.Wednesday’s report showed growth across much of the private sector. The professional and business services sector added 4,200 jobs. Leisure and hospitality grew by 3,800. Construction was up 2,000 jobs.