SAN JOSE, Calif. – Few experts were surprised when Facebook disclosed in its recent IPO filing that its user growth had slowed in the U.S. and Canada. But a deeper look at Facebook’s user numbers shows its growth is also slowing in Europe and Asia, untapped markets seen as vital to its fast-growing business, putting more pressure on the company to assure investors it can keep its revenue and profit expanding.

The Menlo Park, Calif., social networking giant added 45 million regular users – defined as members who accessed Facebook in the past 30 days – in the last quarter of 2011, its smallest increase since spring 2009, and now counts 845 million total users. But even in Asia, where a small share of the population of countries like India, South Korea and Japan are Facebook users, the social network’s fourth-quarter growth was its weakest in six quarters.

Analysts say they are not alarmed about Facebook’s slowing growth, which they say was inevitable, given its overwhelming popularity in many countries. Still, they say it means the company must find new ways to entice people to spend more time on the social network, to show them more advertising while they are there – and perhaps to enter the world’s largest Internet market, China, from which it’s now blocked because of censorship.

With the days of its runaway growth apparently over, Facebook’s future financial success will be more and more tied to “Zuckerberg’s Law” – founder and CEO Mark Zuckerberg’s 2008 prediction that each year, people will share information about themselves at double the rate they had the previous year.

Facebook is “still going to surpass a billion people, even with slowing growth,” said Lou Kerner, an analyst with SecondShares. “So I think the question is, ‘Is Zuckerberg’s Law going to continue to hold?’ “

Compared with the end of 2011, early 2010 was an online gold rush for the world’s largest social network. The number of people using Facebook on a given day grew by more than half a million people with each successive day during the first quarter of 2010. But Facebook added only an average of about 290,000 daily users each day in the final quarter of 2011.

“The worldwide user base is going to slow in growth,” said Rebecca Lieb, an analyst with Altimeter Group. “You can’t have hockey-stick growth forever.”

But the slowdown also means analysts this week are picking though Facebook’s newly revealed “engagement” numbers, which reveal how frequently people come back after they join the social network. They are looking for signs of what has come to be called “Facebook fatigue,” as they try to determine whether the company should be valued as high as $100 billion when its stock begins to trade later this year.

One positive sign for Facebook is a new study by the Pew Internet & American Life Project, which found no evidence of Facebook fatigue among longtime users.