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We keep hearing a lot about economical growth in the media and this answer shows us why it is so important:

Therefore, we have three options:

• A constant increase in unemployment. (Generally feared and loathed.)


• Less time at work per person. (Sometimes impractical. Wastes
  educational resources as people would still have to study and train
  just as much only to produce less. Might reduce people's earnings,
  etc.)


• Constant economic growth to create new jobs and counteract the
  reduced need for manpower caused by technological advancement.

The third option is generally preferred by policymakers, academics, the

public, etc. for the reasons described above, as well as other
reasons.

However, constant economic growth within a planet having finite resources is not possible (it is not sustainable):

Capitalism requires a constantly expanding production and consumption
of goods, which can only be achieved through the increased
exploitation of the planet’s natural resources at an unsustainable
rate. Because of this reality, sustainable development cannot be
achieved without a dramatic reduction in the levels of production and
consumption, which directly contradicts the growth logic that drives
capitalism.

So, I am wondering if how capitalism can deal with this? What if we reach the limits of the environment and economic growth is no longer possible?

Not all parts of the economy consume finite resources equally. In fact, only a minor portion of it is related to the manufacturing of goods. Most of the economy is services by now and, even more related to your question, most of the growth is growth in services.

Manufacturing is on the decline while capitalism thrives.

Source

Growth is growth in services.

Source

In fact, there is plenty potential for services to grow even more. For example, if you live in a western country and you get older, you wish there were more service workers tending to elder people available.

And services are very friendly on the environment. Just an example: Being an Instagram influencer is a very eco-friendly job consuming comparably only few of the finite resources.

Also, some resources are renewable: food, energy, recycled materials (to some extent), wood, .... With regard to manufacturing and production there certainly is a desire to use existing resources as efficiently as possible with this kind of optimization still ongoing.

Yes, saving the planet mostly probably will mean not using more (rather less) of the resources, but that won't be the downfall of capitalism at all. If I would be worried about capitalism, I would worry about ongoing trends in artificial intelligence and automation.

Capitalism is all about maximising profit. Obviously, it can be done in multiple ways, of which reducing the costs is always first and foremost. That includes reducing labor force of an enterprise, which is but one and least efficient, albeit most vilified method.

Most often though capitalism employs new technologies, new materials and new ideas to increase productivity and/or reduce costs. And it can be seen across world and time, that production of anything becomes more efficient, streamlined, less wasteful etc.

There is no reason to assume that the need for natural resources will be constantly increasing ad infinitum without allowing for the possibility of either substitution (humans may develop a technology of converting old plastic bags into new synthetic material stronger than steel, for example) or there will be new sources of resources (i.e. asteroid mining).

In the meantime capitalism's answer is improvement. Not sure how old are you, but I remember when instead of a smartphone one required a huge desktop computer, printer and phone for effective office work. Think on how much technological progress reduced the need for resources going from fifty kilos of various materials to less than 200 grams (which is 250 TIMES) without reduction in productivity (in fact, it increased it dramatically). Is it really hard to apply same logic to almost EVERYTHING (to various extent, of course)?

I would like to (sort-of) challenge one of the premises of the question, namely that resources are finite, and I have a strong and weak version of the challenge.

The Weak Version

I mean, sure, there's only so much matter in the planet, and only so much of it is useful for our current purposes. Whether you think the end is in decades, centuries, or millenia, anyone who can count more than the sum of their fingers and toes can look ahead to the end of the tunnel. But human ingenuity is also a resource, and human creativity is unbounded.

Think about it. Finance? Law? Software? These things are huge sectors of modern economies and they barely consume any physical resources at all. Trilarion's upvoted answer focuses a lot on the service sector, another great example. How much of our resources today exist in the form of intellectual property?

I'm not sure how the theory plays out, but in practice we've already begun to pivot in this direction more and more. N.B. that's exactly what capitalism says should happen: as physical resources become scarcer their value increases and we either have to live with shortages... or pivot to something cheaper.

The Strong Version

Which coexists with the version above: we'll never run out of resources before something else happens.

That something else might be the extinction of the human race, space travel/colonization, digitization of consciousness, whatever. Why aren't we doing more in those spheres now (at least the more positive ones)? Because it isn't economically viable. But again, as physical resources become scarcer their value will increase, until it becomes more viable to strip mine asteroids than continue to deplete the Earth.

There are some holes in the strong version: if agents are allowed to proceed to maximize their utility without regard to negative externalities generated (e.g. a factory can pollute with no consequence) then the incentives are warped. If the ecosystem that sustains life here is wrecked by e.g. AGW then game over. You can probably think of others, but the point is that continuing current trends forward unboundedly is likely selling humankind short.

Capitalism's answer is a global crisis, which weakens world economic powers and redistributes resource consumption. Such a crisis may be either peaceful (like the Great Depression in the US) or non-peaceful (like WWI, for example - when growing economic powers - the German bloc - tried to overpower existing economic powers - the Western world mainly).

So, in fact, there is no such problem in capitalism - its decision is stored inside.

About "growth in services economics, which is unlimited". Let's consider, why it is naive.

The process itself is described very well in @Trilarion's answer. The answer describes so-called "post-industrial" economics.

What's wrong with it?

At first, should notice, that graphics (services percent) are true, but for very special countries from western world, which have such huge services sector (US and UK are most known). You may look at the second graph. We now, that main part of UK GDP is formed by banking sector and services. From the first view, it sounds good - it is a pure service sector, we are just servicing our and foreign banks money, giving credits, so on.

But - of course, there is a 'but' - the financial sector cannot live in a vacuum. When it grows too much over REAL sector, we obtain a financial bubble. It may be a big amount of credits, an exchange bubble, or something else, but what we have? We have an economic sector which growth, in fact, is a fake.

A bright example: Brexit. You may, again, look at the second graph from @Trilarion's answer. The UK have a huge financial sector (result of M. Thatcher politics of 'de-industrializing'). What Brexit means? It means (and Deutsche Bank already have done it), that EU banks will stop to use UK bank services, because of UK is exiting, and retrieve their finances from UK back to homeland. How do you think, this will affect UK bank sector (and, as it is huge, huge part of economics)? It will collapse. You may say - but what's the difference between EU and UK, if both have a financial sector and banks? Yes, both have, but EU do have a big REAL sector (which can be an investment target to the banks, and supports it) and UK does not.

To conclude. That theory - about constant growth, provided with services constant growth - is applicable in a one-hegemony-world, which is globalised. While hegemonic growth its services, which is consumed by others, others, in return, provide something REAL. A bright example is the 'late-Washington' world - before Russian comeback to global powers. But now, as the world becomes multi-polar again, this is not more than a pure theory.