Uploaded September 2021 | Updated September 2026, 1 week ago
e^x limit for real numbers: youtu.be/HM-kwHR4VO4
Euler's formula for the complex exponential can be proved in many different ways, such as with Taylor series. Here we find Euler's identity using a limit equation for e^x and the polar form of complex numbers!
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
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Music: OcularNebula - The Lopez
e^x limit for real numbers: youtu.be/HM-kwHR4VO4
Euler's formula for the complex exponential can be proved in many different ways, such as with Taylor series. Here we find Euler's identity using a limit equation for e^x and the polar form of complex numbers!
Calculus Problems playlist: youtube.com/playlist?list=PLug5ZIRrShJGFne7YhMi-4eYsUKzkITao
Subscribe to see more new math videos!
Music: OcularNebula - The Lopez
![Matrix Invertibility With F[x]-Modules](https://i.ytimg.com/vi/xnZ2d6y8JmA/mqdefault.jpg "Matrix Invertibility With F[x]-Modules
Characteristic polynomial explanation: https://youtu.be/jCt6mR3QtPk
Ring & Module Theory playlist: https://youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
In this video, we use F[x]-modules to prove that a matrix is invertible if and only if its determinant is nonzero. Another example of how F[x]-modules can be applied to a ton of things in linear algebra!
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Music: OcularNebula - The Lopez")
