Apple Inc. gave a quarterly sales forecast for the critical holiday shopping season that fell short of some analysts’ estimates, showing there may be a limit to the growth it can squeeze out of the iPhone.

Fiscal first-quarter sales will be $75.5 billion to $77.5 billion, the Cupertino, California-based company said Tuesday in a statement. Analysts on average projected $77.1 billion, according to data compiled by Bloomberg. While revenue will be up at least slightly from a year earlier, growth is far below last year’s 30 percent jump during the holiday rush. After reporting sales rose more than 20 percent for the past four quarters, analysts predict Apple won’t top 10 percent growth through at least 2017.

Apple’s sheer size means growth is tougher to achieve. As iPad tablet sales decline and adoption of Apple Watch remains modest, the company’s expansion has become increasingly dependent on demand for new iPhones. After the long-anticipated debut of bigger-screen handsets in 2014, sales reached a record 74 million units during the holiday quarter last year, leading investors to question how much higher sales can go.

“The law of large numbers is working against them as they get bigger,” said Alex Gauna, an analyst at JPM Securities, who recommends buying the stock. “It gets harder to show growth.”

Apple shares fell less than 1 percent to $114.55 at the close in New York.

Fourth Quarter

Apple included its holiday outlook in results for the fiscal fourth quarter, which ended on Sept. 26. For that period, the company posted net income of $11.1 billion, or $1.96 a share, while sales rose 22 percent to $51.5 billion. Analysts had predicted earnings of $1.88 a share on sales of $51 billion. The company had $205.7 billion in cash and investments on its balance sheet at the end of the quarter. Gross margin, a measure of profitability, widened to 39.9 percent.

Apple’s success largely hinges on the iPhone. The company sold 48 million handsets last quarter, up 22 percent from a year earlier and shy of analysts’ average prediction of 48.5 million shipments. The device generated sales of $32.2 billion, making it bigger than Microsoft Corp. and Facebook Inc.’s quarterly businesses combined. Introduced on Sept. 25, the newest models - - iPhone 6S and 6S Plus -- come with an improved camera, faster processor and new 3D Touch screens.

Smartphone sales have helped Apple make up for falling demand for the iPad. Shipments of the tablet fell 20 percent to 9.88 million units in the quarter, the seventh straight decline. Apple didn’t break out unit sales for Apple Watch, but analysts estimated the company sold 3.8 million in the quarter. Mac purchases rose to 5.71 million, compared with an average projection of 5.6 million.

For the all of fiscal 2015, Apple reported total sales of $233.7 billion. Net income grew 35 percent to $53.4 billion for the year.

Two Approaches

Two schools of thought have emerged around Apple’s business. To skeptics, the company has largely run out of new geographic regions to introduce the iPhone, and without new hit products, it’s reaching the limits. Others argue Apple has room to grow because it can still gain market share in the global smartphone market, and that add-on services such as Apple Music and Apple Pay will boost revenue.

“The debate is going to rage on,” said Gauna, who sides with those optimistic about the company’s future.

Chief Executive Officer Tim Cook is banking on China to add new customers. Last quarter, the company generated $12.5 billion in revenue in the region, including Taiwan and Hong Kong, almost double the year-earlier sales, yet down 5.4 percent from the third quarter. Cook, who visited China last week, has said the world’s most populous country will eventually be Apple’s biggest market.

Apple refreshed its product lineup to take advantage of the year-end holiday rush. The company generates more revenue and profit during this period than any other quarter in the year. In addition to the new iPhones, the company remodeled the Apple TV set-top box and updated the iMac with sharper screens.

Apple’s stock has fallen more than 12 percent since its last earnings report in July amid concerns about slowing growth. The stock’s performance shows the fickle nature of Wall Street. While Apple is the world’s most valuable company, and has a balance sheet that other companies envy, many investors make decisions largely based on the outlook for future growth. Apple faced a similar challenge in 2013, when sales slowed amid competition from Samsung Electronics Co.

“They get graded on a curve,” JMP’s Gauna said. “In technology, there are the quick and the dead, so Apple needs to be on the side of the quick.”