Alan Najat
New member
- Joined
- Jun 11, 2024
- Messages
- 9
hey, I've gotten really confused over something even though it is kinda simple I guess, once I saw a question that was asking if wether f(x)=x^(4/5) has a vertical tangent or not, I believed it had, but for certainty I asked Chat GPT and he said it has it, so I answered the question and said it has it, but a teacher that has master's degree in mathematics said no, it doesn't have a vertical tangent, it only has Cusp, I was confused, after several months I got back to the same problem but this time I checked the CALCULUS book as a source to know for sure what is it, 

in these images, example number three shows an example similar to the one I provided, which is f(x)=x^(2/3), at first it says that it doesn't have a vertical tangent, but then it says "In the light of the two preceding examples, we extend the definition of tangent line to allow for vertical tangents" which then provides a new and broader definition that allows the function to have a vertical tangent, I argued with Google Gemini and now I'm convinced that it does actually have vertical tangent, here is the full link for my conversation with Google Gemini
my teacher's argument was that the limit of the derivative must be +infinity or -infinity in both sides at the same time, but in this example the signs of the infinities are different from the both sides, so he said it doesn't have vertical tangent, I would be great full if you told me and re assured me wether this function have a tangent at 0 or not, if yes, is the tangent vertical?
thanks.




my teacher's argument was that the limit of the derivative must be +infinity or -infinity in both sides at the same time, but in this example the signs of the infinities are different from the both sides, so he said it doesn't have vertical tangent, I would be great full if you told me and re assured me wether this function have a tangent at 0 or not, if yes, is the tangent vertical?
thanks.