Exercise 1: Precise Definition of a Vector Function Limit

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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!

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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