Fundamental properties of nature, like the finite speed of light, preclude infinite growth. To maintain 2% annual growth in human energy consumption, in just a few thousand years we'd need to have built Dyson swarms around every star in our galaxy (and beyond). Faster-than-light travel becomes necessary to sustain that growth rate in less than 2600 years.

The BBC asked a bad question and got an obvious answer. (Maybe the badness and obviousness aren't apparent up front if you don't have a physical science background.)

Asking if fixed-percentage economic growth can continue forever is rather like asking if Moore's Law can continue forever. No, neither can continue forever. That doesn't mean that things get bad after the growth phase. Somehow both optimists and pessimists conflate "the end of growth" with "the end of prosperity." That's ridiculous, IMO. The median citizen is much better off in low-economic-growth (and negative population growth) Japan than in rapidly growing Bangladesh.