Limits & Continuity

Question 1

Theorems of Limits

Answer 1

Question 2

Squeeze Principle

AB Equatoins to Memorize

Answer 2

Two Equations to Memorize:
A / (A+B) ≤ 1

|AB| / (A² + B²) ≤ 0.5

Question 3

Two-Path test for Nonexistence of Limits

Answer 3

It limit approaches two different values from two different angles, the function has no limit at (a,b)

Question 4

Prove Existence of Limit

Answer 4

1. Substitute ( a,b ) into ( x,y )
2. Use Approximation ( Squeeze Principle)

Question 5

Continuity

Answer 5

A function is continuous at a point if:
1. f ( a,b ) exists
2. lim f ( x,y ) as ( x,y ) approaches ( a,b ) exists
3) f ( a,b ) = lim f ( x,y ) as ( x,y ) approaches ( a,b )

Note : You can just substitute ( a,b ) into ( x,y ) for limits of continuous functions

Question 6

Continuous Extension

Answer 6

When a function has a “hole” add an extension to not have a hole.
Often using the squeeze principle or Canceling factors