If there is no upper limit on population growth, you also need no upper limit on your rate of expansion.
Suspended animation could work, but now you've just admitted that you're putting a limit on reproduction, but even that ends when those people wake up and arrive somewhere.
Even allowing all that, "we'll solve this unsolved problem by solving another unsolved problem!" does not inspire a lot of confidence.
Immortality will reduce reproduction rates intrinsically; 500 years ago it was common to be married off and breeding before you were 15, now you're 'having kids young' if you start before 25, how long will people wait when they live 100s of years?.
Improving global standards of education and quality of living also slow down reproduction. Countries scoring highest in both of these metrics are often fairly stable in population, some even depend on immigration to maintain current levels.
Residual growth after these two factors are taken into account is low enough that the odd colonisation attempt or newly built orbital habitat will be ample space. And this is without even considering that we could end up living as uploads in virtual worlds long before these things become possible.
And the issue I'm referring to with food production is the order and distribution of the diffusion of the technology. In Europe, the agricultural revolution preceded breakthroughs in medical technology, and it worked out great. However, the rate of diffusion of medical technology to the east was greater than the diffusion of agricultural technology, which is what caused catastrophic results. Yes, they could have produced enough food, but because the rate of diffusion of the technologies was different, they didn't. We should apply this historical lesson.
In Tragedy and Hope, Carroll Quigley argues:
> importation of an element of material culture from one society to another is helpful to the importing society in the long run only if it is a) productive, b) can be made within the society itself, and c) can be fitted into the nonmaterial culture of the importing society without demoralizing it.
For longevity technology, a) is perhaps true but it's not clear that b) and c) would be for most societies. There's a calculus here worth getting right, otherwise longevity technology could end up doing more harm than good.