Exercise 1: Precise Definition of a Vector Function Limit

In this video I go over the precise definition of a vector function limit and prove it in 2 different ways. The 1st solution involves assuming the vector exists, and then using the basic definition of a vector function limit as well as the 3D distance formula, to prove the difference inequalities of the precise definition. The 2nd solution is the reverse: it involves assuming the precise definition difference inequalities and then working backwards to prove the vector function limit exists. Lastly, I show that the precise definition of a vector function limit in 3D can be visualized as a sphere which gets indefinitely smaller as the interval shrinks. This is probably the only in-depth precise definition of limits for vector functions anywhere in the internet, so buckle up!

#math #vectorfunctions #limits #science #calculus

Timestamps

• Exercise 1: Precise Definition of a Vector Function Limit: 0:00
• Recap on limits for real-valued functions: 1:23
• Solution 1: Assuming the limit exists: 5:15
  • Applying Definition 1 for limit of a vector function: 7:49
  • Equating components of vector function and the limit to obtain 3 real-valued limits: 9:40
  • Summing up the inequalities of the components: 13:56
  • Using the 3D distance formula for the difference inequality: 15:44
  • Difference is less than or equal to square roots of each part separately: 17:50
  • Difference is less than ε, this proves the precise definition of the limit: 19:50
• Recap on the Distance formula in 3D between 2 points: 21:40
• Solution 2: Assuming precise definition first and working backwards prove our limit: 22:28
  • Expanding the difference inequality as their vector components and applying the distance formula: 23:55
  • Square both sides to reveal each component difference is less than ε: 25:30
  • Applying the definition of limits of real-valued functions for the components: 28:42
  • Applying Definition 1 for Vector Function Limits proves our precise definition for the limit: 29:40
• Visualizing the Precise Definition of a Vector Function Limit: 31:50
  • Precise definition plots a sphere around the limit that shrinks as the interval shrinks: 38:15

Notes and playlists

• Summary: @mes/re-leothreads-2bsgqbqqj
• Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GwdMY-E9_4LsmYmO-IxcoX
• Notes: @mes/omhbcxkw
• Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

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