Access to Claude

Hi all. I am a paid Cursor subscriber and learning new things every day and one thing I jhave found is that I could do with using something like chatgpt alongside the cursor AI but I keep running out of requests really quickly unless I get a pro account. I found the Claude AI website and thought as I have Claude in Cursor, I would be able to have the same access i do through cursor but it does not look like that is the case.

Does anyone know if we should be able to use Claude AI website with unlimited requests as we can do in cursor?

I have tried doing certain things with Claude in cursor and it cannot generate what I want as it is quite long winded, albeit veery straight forward.

Is there maybe a way in Cursor to just use Claude separately from creating an app?