DISCLAIMER: I am not a lawyer.
Yes, you can use Extjs on your website without asking permission (unless you distribute it). There are many questions about the GPL, so I won’t go into details.
If you are encoding an application for a client, it is intended only for use by this company. You do not need to disclose your source code. On the other hand, if you are creating a web application that you sell (or distribute) to several companies.
However, they offer great products. Therefore, do not hesitate to support them by buying a license.
dan-klasson
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