Yes, the human race is quickly approaching overpopulation.
No, there has been many population limits claimed in the past, all of which were solved with technology, all limits in the future will be past with technology.
We already are vastly overpopulated. At our current rate of consumption, we would need 2 Earth's to maintain our population and lifestyle. We consume the Earth's yearly production in August of every year, creating an ongoing resource deficit. We are driving full speed into a brick wall that lies squarely within the 21st century. It is just common sense and basic math.
I am just waiting for Elon Musk to make developments on setting up Mars bases. If we continue staying in Earth, we are not only in danger of degrading the earth as we already have (Oil Spills, deforestation, melting of ice caps, increasing temperatures due to air pollution) , the human race is also more susceptible to an extinction event like a meteor strike on Earth. So whatever may be the case. we are still bound to become multi planetary species in the next 40-50 years.
We're already greatly overpopulated. The Earth can sustain four billion people without any trouble and at the current resource consumption rate. Our population is approaching eight billion, double of what the Earth can hold. We need to get a handle on this, and fast
Humanity can adjust the carrying capacity that the Earth would normally allow through better fertilizers on crops, genetically modifying crops, better use of land, and using other resources more carefully. The places that use the most resources (US, Europe, China) also have very stable growth rates at around 2% and therefore are maintaining an equilibrium as well.
No. That is, we are not approaching a point where this overpopulation will destroy us. Why not? Well, there are two reasons. First of all, population growth has actually slowed, and growth is expected to have decreased dramatically by the time we reach a population of about 10 billion humans. Second, technology is helping to compensate for population increase. For example, in the year 1800 human civilization would be perhaps incapable of survival with a population of 7 billion. Today, things are strained in many places, but we are getting by with such a high population. There are many upcoming technological innovations that can help us compensate. For instance, technology is progressively becoming more green and efficient. We will within 2 decades or so be extracting resources from resource-rich asteroids, which would mean a huge increase in the amount of available resources. Within 100 years it's probable that human consciousnesses could be translated onto an electronic format and humanity could exist mainly in cyberspace, which means that we'd be using up way resources. Overpopulation may cause problems, but it won't be the end of us.
What I have to say about this, others have already said and I've also said in other parts of the site.