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Courses / Calculus I / Calc-1 Discussion 01

3 Types of Discontinuity

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Discussion Topics covered: 3 Types of Discontinuity Graphically / Multiplying by the conjugate and algebraic manipulation

Part A: Classification Matrix (~45 mins)

Instructions: Before solving, classify each problem, identify the governing rule or theorem, and justify your classification using visual clues.

graph TD
    A[Identify Visual Clues] --> B{Form of the function?}
    B -->|Products/Quotients| C[Product/Quotient Rule]
    B -->|Composite functions| D[Chain Rule]
    B -->|Trig/Exponential| E[Special Derivative Rules]
Problem Classification Identifying Rule/Theorem Visual Clues / Justification
1
2
3
4

πŸ“… Problem 1

Evaluate the limit graphically and algebraically: lim⁑xβ†’2x2βˆ’4xβˆ’2\lim_{x \to 2} \frac{x^2 - 4}{x - 2}. Classify any discontinuities.

Solution to Problem 1

Step 1: Simplify algebraically. Notice that the numerator is a difference of squares:

x2βˆ’4xβˆ’2=(xβˆ’2)(x+2)xβˆ’2=x+2forΒ xβ‰ 2\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for } x \neq 2
Step 2: Evaluate the limit.
lim⁑xβ†’2(x+2)=4\lim_{x \to 2} (x + 2) = 4
Since direct evaluation yields 0/00/0 but the limit is finite, this is a removable discontinuity (hole) at x=2x = 2.