I disagree with Madara's answer claiming that there are only two possible ways to answer identification requests. This seems like a drastic oversimplification, to the point that it's factually incorrect. Most of this post is intended as a reply to that line of thinking. For my answer to this question, see the last paragraph.
To put it simply, I rarely use Google image search simply because it's rarely the best tool for the job. To make an analogy to programming, if
for loops are the only form of loops you've ever used, you'd think they're sufficient, but better programmers know that there are lots of different types of loops which have different use cases. I'm sure that for some users here, Google image search is the only tool they use regularly to ID images, but with that you're really only getting the easy ones, and often times not getting as much as you could be.
To give several examples from my history where Google image search was either unnecessary or unhelpful (feel free to skip to the conclusions if you prefer):
In this question, Google image search turned up 0 relevant results at the time I was searching. It didn't answer the question. However, there are steps you can use to answer it. First, looking at the style of the image, it's likely that this is not from an animated medium. The quality is too high and the style is different. It looks like a typical VN CG.
In fact, I'd seen images of the same characters before, though only a couple more pictures. But based on that, I was able to figure out through Anime Characters Database that the girl was this character. Browsing a few CGs it was enough that I was convinced it was the same. Later I was able to go back and confirm this and identify which VN in the series it was. Google image search literally never helped with this.
For this one, you could have gotten the right answer through Google...if you cropped the image to just the upper right panel and then waded several pages of results (all in Japanese) before finding the artist's pixiv profile (also in Japanese), and then browsed that to find the original image.
Instead, it was faster to just do an ordinary Google search for "leben nicht site:pixiv.com". This took me to a different image of the same character, but that pointed me back to the original artist. Still needed to know Japanese, but that's unavoidable in answering this sort of question. Google image search was just not helpful, though you could have made it work if you were desperate.
This one is pretty well documented on my answer itself, and not a great example since I didn't even succeed, but it's at least evidence that Google isn't the only tool here. The first thing to do was to do a Google image search to find the highest resolution copy of the same image, but that's where Google's usefulness ends. The image was simply on too many "Free Anime Wallpaper" sites to show up with anything remotely useful. However, the full resolution image confirmed suspicions from the small image that this was photoshopped. Incidentally, if you thought to flip the image and then went to page 24 (for me) of the Google image search results, you would have found a link that would eventually lead to this post on e-shuushuu. More specifically, it took me to an e-shuushuu search page which was cached, and going a couple hundred images in you would have gotten to this image. I only discovered this after I'd already found that image page through other means. This sort of thing is often highly dependent on your search history and location and changes regularly, so it's not a good idea to rely on it.
To find the original image, I went on an image board and just searched for images of girls with flutes and purple hair and head ornaments. After a few dozen images, I managed to find the original one (posted in the answer). That gave me the artist's name. Cross-referencing a different image board (which had other images by the artist but not this one) was enough to get me to the artist's Pixiv page and Wikipedia page. Exhaustively checking that it wasn't from anything listed on Wiki was enough to confirm my suspicions based on the image's style and resolution that it wasn't from any anime, though I still wasn't able to locate the original. But this was at least enough to write an answer, and for the reasons I stated in my answer going farther would be hard here.
In this case, Google image search didn't give anything useful. However, I happened to already know the source in question. Based on that information, I was able to locate the original uncropped version of the picture. Google image search for the uncropped image did give me a few nice facts to put in my answer, but they weren't necessary to answer the question. Even if I hadn't known the source, I could have still found it just based on what I could see from the picture by going to an image board. Note that for me it now shows the full image under the "related images" section, but this was not true at the time and may be partly due to my answer.
This is one example from several years ago (before this site existed), but I figured I'd share it since it's quite different from anything else here. One of my friends had a hard copy of a certain piece of fanart of a character. We knew the character, but not the artist. I scanned his image, but Google didn't give any results. But since we knew the character, I could just download every (high quality) image on Pixiv with that character with a program. I then ran that archive together with the scanned copy through a duplicate checking program, and sure enough it found the duplicate which pointed back to the original artist. Sure, anyone could have done that, but it's at least evidence that Google isn't the only useful tool.
In at least five other cases, I've simply had no use for Google image search, though it may have given the correct answer. For instance, in this question I knew the answer immediately. It didn't require Google. I eventually did a Google search, but the only additional thing I learned from it was additional places that the image was used. In this case, I knew the answer without any Google searching. It turned out that appropriately cropping the image, one could get some relevant results with Google. In this case, I again answered without Google image search, and later used Google to figure out exactly what page of the manga the image was from. Sure, in all of these cases, Google alone could have given the answer, but it was simply faster to just know it. Here's an example where I answered in under a minute; if you think you could beat that by Google image searching, you're almost certainly mistaken. I could give more examples to this effect, but I don't really think there's room for debate at this point.
So no, it isn't just a matter of "Google or nothing". There are plenty of other ways to identify a series. If you didn't realize this, that's probably why I was able to answer these questions and you weren't. But I'm far from the only user here who has answered ID requests which didn't appear on Google. I'm quite certain that this image had no results on Google when it was asked. I know because I tried it myself. The same is true for at least most frames of this gif. That doesn't mean they aren't answerable; the answers were found.
Also, it's worth pointing out that if your idea of a good answer is just "Google and copy the title", that's not what a good answer means in this case. There's plenty more that you can say about an image apart from just the title of the series. Some of these are things that you can find via Google (e.g. characters, artist). Some are things you can't (e.g. at what time a screencap occurred in an anime). But in general, answers here should be informed and contain as much information as possible, not just the stuff Google gives you for free. That's true not just for ID requests, but for all questions on this site. If the best you can do is copying from some other site, it's not a very good answer. We should be striving to do better than other sites, not copy their content.
As for asking the OP to provide more information, e.g. where they found the image, this is sometimes of use. But often it's unnecessary. If they found it on a friend's facebook wall, I'm not going to try to get in contact with their friend to ask what it's from. I doubt anyone else is either. We have better methods of IDing images than relying on someone's friend. Not that I mind such content being added, but if there's already an image present, I don't see any reason to mandate it. It might help in 20% of cases, but 80% of the time it's basically irrelevant. That doesn't seem like a good thing to determine the fate of questions on.
We should continue to allow these questions because there's simply no reason to ban them. If they're easy to answer with Google image search, then answer them. If you don't feel like it, then don't. If you think that the OP didn't do enough research before asking here, downvote. But there's no good reason to close them. And especially for the images where Google fails, these absolutely shouldn't be closed. Such questions are certainly challenging to answer, but that doesn't mean they're unanswerable. They aren't "unclear" at all; you know exactly what the OP wants, but you just don't know the answer (because Google doesn't know the answer and apparently you're incapable of beating a machine). They have answers, and if people here are smart enough and knowledgeable enough we'll eventually find most of them. Closing is for questions which will just not work in this format, either because they aren't questions at all or because they're too subjective. Closing is absolutely not for questions just because they're difficult. If we're going to start closing hard questions, we might as well ban every anime with fewer than 100,000 viewers on MAL, because the chance that anyone here will know the answer is low.