SAN FRANCISCO — Michael Dell and Meg Whitman may be business competitors, but they can each claim a technology industry superlative. One has overseen one of the largest mergers in the tech industry. The other has engineered its biggest breakup.

Different as they seem, both deals were in reaction to tough competition and declining markets. And so far, neither has provided a clear way for how a storied giant can survive changes like the rise of so-called cloud computing that are undermining the way they operate.

On Wednesday, Mr. Dell, who 32 years ago started a personal computer business from his college dorm room, officially completed his giant $67 billion merger with EMC, a data storage company with interests in software and security. He also changed the name of his operation to Dell Technologies.

Later in the day, Ms. Whitman, the chief executive of Hewlett Packard Enterprise, announced a sale of most of her company’s software assets to a British outfit called Micro Focus for about $8.8 billion. H.P.E. also announced higher earnings in the company’s most recent quarter, but a decline in revenue from the same period a year ago.

Shares of H.P.E. moved down about 2 percent in after-hours trading Wednesday evening.

H.P.E. itself is a $36 billion computing giant created last year from half of Hewlett-Packard, a record separation that ended one of Silicon Valley’s most iconic companies and neatly mirrored Mr. Dell’s move. The other half of the old company, which specializes in PC and printer sales, is now called HP Inc.

With a constant mantra of building shareholder value, Ms. Whitman, who a year ago talked about growth strategies for H.P.E., has sold off big parts of the empire she took over five years ago. Last May, this involved selling off her services group, which employed 110,000 people. In an interview, Ms. Whitman said that after that sale, by the end of October, H.P.E. would have between 50,000 and 60,000 employees.

“We have to understand our place in the world,” she said. “There are elements of our world that are shrinking.”

Mr. Dell’s move in the other direction has meant taking his company private, in the process taking on $33.4 billion in long-term debt at relatively little personal risk. He owns 70 percent of Dell Technologies, a company he is counting on to throw off a lot of cash to quickly repay the debt, even as the overall business computing industry shrinks.

If his gambit succeeds, within a few years he could multiply by several times his personal fortune, already estimated at $18.5 billion, according to Bloomberg.

Dell Technologies starts life with revenue of $74 billion and 140,000 employees, but in an interview on Wednesday Mr. Dell declined to say what those numbers might look like in two years.

“Things are going well on a relative basis,” he said. “Market consolidation is definitely occurring.”

With companies like Amazon Web Services and Google providing cloud computing services, corporations increasingly rent the use of computers they used to buy from Dell and H.P.E., for a lot less money. Other cloud companies, like Salesforce, rent software this way, something A.W.S. and Google also do.

Along with the rise of mobile computing, cloud computing is radically altering the computer industry. According to IDC, global demand for traditional data storage fell 7 percent in the first quarter of this year. Server revenue fell 3.6 percent globally. Second-quarter PC sales were down 4.5 percent. In a market like that, big companies like H.P.E. and Dell can only hope for a bigger slice of a shrinking pie.

Compare that with Amazon’s fortunes. Since A.W.S. first disclosed its earnings in April 2015, Amazon’s stock has nearly doubled, and despite a 40 percent increase in the tech-heavy Nasdaq stock market, shares of almost all the older business technology companies have barely budged.

Cloud companies now get about 10 percent of all spending on business computing, and they are growing at a 30 percent annual clip, said Toni Sacconaghi, an analyst with Bernstein Research. “That’s 100 percent of the growth, maybe a little more” of the whole business technology sector, he added. “A year ago Meg talked about growth, but in the past six months it’s all been about capital allocation and return to shareholders.”

Compared with many in the old guard, the H.P.E. and Dell strategies may be doing well. At least for now.

Shares of H.P.E. have gained about 30 percent in the last year, largely in the last two months, as rumors spread that besides selling services and buying back shares, Ms. Whitman would sell the company’s laggard software division.

Pre-EMC, Dell was known mostly for PCs and computer servers. While the financial information released by the private company is limited, in the three months that ended April 29 it had revenue of $12 billion, down 2 percent. Operating income was higher, Dell said, using nonstandard accounting.

Selling computers still throws off a lot of cash, which is good for paying debt. Mr. Dell said that Dell Technologies threw off $3.6 billion just in its most recent quarter. EMC also owns a software business, called Pivotal, which Mr. Dell is expected to take public soon.

“He’s betting that size and scale will help him gain share, not that the market will be great,” Mr. Sacconaghi said.

Computing, once the province of a few business tasks and consumer activities like video games, has with cloud computing attached itself to much of the world, making it possible to connect phones, tablets and sensor-equipped machines to huge computing resources.

Both Dell and H.P.E. make the computers and related devices that amount to the infrastructure necessary to this new economy. Increasingly, however, the profit margins can be found above the computing guts, in particular with artificial intelligence technology that helps companies scour data to make smart decisions. Both Microsoft and IBM have invested heavily in both cloud computing and A.I.

Mr. Dell said that he had spent $12.7 billion in research and development in the last three years and is gaining some A.I. technology in the EMC deal.

In turn, when asked about its innovations, H.P.E. noted changes to its server and storage businesses, along with the acquisition of companies like Silicon Graphics, a high-performance computing company that peaked in the 1990s.