Limits
In 2D, you could only approach from left or right. In 3D, you can approach from infinite directions.
Introduction
For a limit to exist, the function must approach the same value along every possible path to .
The Two-Path Rule:
If you find two different paths that give differents limits, the limit does not exist.
Continuity & Factoring
If a function is continuous at , just plug in the point!
Technique 1: Direct Substitution
Use for polynomials, exponentials, sines, cosines (where defined).
Technique 2: Factoring
Use for forms that look algebraic.
Visualizing Paths
Interactive: Approaching the Origin
If approaching along red gives and approaching along blue gives , and , the limit DNE.
Worked Examples
Example 1: Does Not Exist (Lines)
Show does not exist.
- Along -axis ():
. - Along -axis ():
.
Since , the limit does not exist.
Example 2: Requires Parabola Path
Show does not exist.
Along any line (parabola opening sideways):
The limit depends on !
If , limit is . If (y-axis), limit is .
Limit does not exist.
Example 3: Using Polar Coordinates (Advanced)
Evaluate .
Switch to Polar: . As , .
Since is bounded, as , the whole expression goes to 0 (Squeeze Theorem).
Limit = 0.
Practice Quiz
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