Exercise 1: Precise Definition of a Vector Function Limit

Math Easy Solutions

math vector functions limits science calculus

description

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