Predicting that far ahead is probably futile. Any number of things could wipe your descendants off of the earth before then. Right this second, I have two descendants and got myself snipped, so a lightning strike at the wrong time could do it. In the next thousand years, it seems unlikely that we won’t have at least one global pandemic and that can wipe out your descendants pretty quickly. A war can take them out if it’s at the wrong time and place. Plenty of Syrians alive 20 years ago have no descendants left. With the trend for smaller families, my guess is the biggest threat to your descendants though is just them not bothering to reproduce. Like I said, I have 2, current trends in first world countries put childlessness somewhere around 20 percent and likely to rise, so with 2 kids, it’s probably a 1 in 20 chance that my line ends with them. Due to their race, expected education level and expected income, the chance is likely much higher. Smaller family sizes and decreases in fertility make it so that having no descendants due to failure to procreate a reasonable outcome.
Of course, this isn’t really a question about why your particular genes will cease to be, but rather is the result of climate change going to be human extinction. The answer is… probably not. Humans are really awesome in our ability to adapt and survive. We pretty much can survive and even thrive in almost any ecosystem and that’s even before the introduction of more modern technology. I mean look at the Tuaregs. They survive in the freaking Sahara. Even if we completely have runaway global warming that turns most of the world to barren desert, it’s likely that people will find a way to survive.
That doesn’t mean that we can just smile and burn fossil fuels until there are none left. Most of us want to do more than simply survive. I’m a mountain boy and you could plop me down in the Alleghenies and I could probably make a pretty good run at survival, doesn’t mean I’d like it or I’d wish it upon future generations. I like having food security and fresh water and nice trees and not horrific diseases and deadly storms. Climate change is going to make things harder and that’s the real bottom line. People will die because of it and a fairly decent chance that it will be large amounts of people. Like I said earlier, the Tuareg can survive in the desert, but pre-modern era, it was not a fun existence, with mass starvations not uncommon and a requirement to war with people in better climates for resources. That’s not a life that most of us would want for our hypothetical descendants. So, will climate change make humanity extinct? Probably not. Will it make our lives harder? Definitely so. Will it make our lives significantly harder to the point where survival becomes more of a priority over what we’re doing now? For some people, that’s very, very likely and it’s not out of the realm of probability that that could be so for the majority of humanity.