Functions, Limits & Continuity: Graphs, Types & Behavior

TAKEAWAYS:

• Functions map inputs to outputs uniquely; their graphs reveal behavior (linear, quadratic, piecewise).
• Limits describe a function’s trend near a point (even if undefined there), using ${\lim }_{x\to a}f(x)=L$.
• Continuity means no jumps/breaks in a function’s graph; removable vs. essential discontinuities differ.
• Piecewise functions model real-world rules (e.g., tax brackets, delivery fees).
• Graphical analysis (holes, asymptotes, jumps) predicts system stability (e.g., stock prices, queues).

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1. Functions: Definition and Classification

A function $f:X\to Y$ assigns exactly one output $y\in Y$ to each input $x\in X$. Notation: $y=f(x)$.

Types of Functions

Worked Example 1: Classifying Functions

Problem: Classify $f(x)=\frac{x^2-1}{x-1}$ and sketch its graph. Solution:

1. Simplify: $f(x)=\frac{(x-1)(x+1)}{x-1}=x+1$ for $x\ne 1$.
   • Hole at $x=1$: $f(1)$ is undefined, but ${\lim }_{x\to 1}f(x)=2$.
2. Graph: A line $y=x+1$ with a hole at $(1,2)$.

Real-World Tie: eSewa’s transaction fees use piecewise rules:

• $f(a)=0.5\%$ of amount $a$ if $a\le \text{NPR }10,000$,
• $f(a)=1\%$ if $a>\text{NPR }10,000$. This is a piecewise function with a "jump" at $a=10,000$.

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2. Limits: Intuition and Formal Definition

A limit describes the trend of $f(x)$ as $x$ approaches a value $a$, even if $f(a)$ is undefined.

Key Concepts

• Left-hand limit (LHL): ${\lim }_{x\to a^{-}}f(x)$
• Right-hand limit (RHL): ${\lim }_{x\to a^{+}}f(x)$
• Two-sided limit exists if LHL = RHL.

Worked Example 2: Evaluating Limits

Problem: Find ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$. Solution:

1. Direct substitution fails (denominator = 0).
2. Factor numerator: $\frac{(x-2)(x+2)}{x-2}=x+2$ for $x\ne 2$.
3. Limit: ${\lim }_{x\to 2}(x+2)=4$.

Real-World Tie: NTC’s internet speed limits near capacity:

• As users approach 100% bandwidth, speed $f(u)$ drops toward 0.
• ${\lim }_{u\to 100\%}f(u)=0$ (speed → 0), but $f(100\%)$ is undefined (no service).

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3. Continuity: Definition and Types

A function $f$ is continuous at $x=a$ if:

1. $f(a)$ is defined,
2. ${\lim }_{x\to a}f(x)$ exists,
3. ${\lim }_{x\to a}f(x)=f(a)$.

Types of Discontinuities

Worked Example 3: Testing Continuity

Problem: Is $f(x)=\{\begin{array}{ll}{x}^2& x\le 1\\ 2x-1& x>1\end{array}$ continuous at $x=1$? Solution:

1. Check $f(1)$: $f(1)=1^2=1$.
2. LHL: ${\lim }_{x\to 1^{-}}{x}^2=1$.
3. RHL: ${\lim }_{x\to 1^{+}}(2x-1)=1$.
4. Conclusion: LHL = RHL = $f(1)$ → continuous.

Real-World Tie: Khalti’s transaction processing:

• If $f(t)$ = transaction time for amount $t$, it’s piecewise continuous:
  • For $t\le \text{NPR }5,000$, $f(t)=2$ seconds (constant).
  • For $t>5,000$, $f(t)=2+0.001t$ (linear).
• Discontinuity at $t=5,000$: Check if LHL = RHL = $f(5,000)$.

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4. Piecewise Functions and Real-World Modeling

Piecewise functions define different rules for different intervals, like:

• Tax brackets (Nepal’s income tax):

  flowchart TD
    A["Income ≤ NPR 500,000"] -->|"0%"| B["Tax = 0"]
    A --> C["Income > 500,000"] -->|"10%"| D["Tax = 0.10 × (Income - 500,000)"]

• Pathao’s ride pricing:

  flowchart TD
    A["Distance ≤ 1 km"] -->|"NPR 20"| B["Base fare"]
    A --> C["Distance > 1 km"] -->|"NPR 10/km"| D["Fare = 20 + 10 × (Distance - 1)"]

Worked Example 4: Piecewise Limit

Problem: For $f(x)=\{\begin{array}{ll}3x+1& x<2\\ 4-x^2& x\ge 2\end{array}$, find ${\lim }_{x\to 2}f(x)$ and check continuity. Solution:

1. LHL: ${\lim }_{x\to 2^{-}}(3x+1)=7$.
2. RHL: ${\lim }_{x\to 2^{+}}(4-x^2)=0$.
3. Limit does not exist (LHL ≠ RHL).
4. Discontinuity at $x=2$ (jump).

Real-World Tie: Nepal’s traffic rules at Thapathali junction:

• $f(t)=$ waiting time for vehicles:
  • $f(t)=0$ if $t$ is green light duration,
  • $f(t)=30$ seconds if $t$ is red light duration.
• Discontinuity at light changes (sudden jumps in $f(t)$).

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5. Graphical Analysis of Limits and Continuity

Use graphs to identify:

1. Holes (removable discontinuities),
2. Jumps (LHL ≠ RHL),
3. Asymptotes (infinite discontinuities),
4. Corners (non-differentiable but continuous points).

Worked Example 5: Analyzing a Graph

Problem: Sketch $f(x)=\frac{x^3-8}{x-2}$ and analyze continuity. Solution:

1. Factor: $f(x)=\frac{(x-2)(x^2+2x+4)}{x-2}=x^2+2x+4$ for $x\ne 2$.
2. Graph: Parabola $y=x^2+2x+4$ with a hole at $x=2$.
3. Limit: ${\lim }_{x\to 2}f(x)=4+4+4=12$.

Real-World Tie: NEPSE stock prices during splits:

• If a stock splits 2:1, its price $f(t)$ has a hole at the split date.
• Example: If stock was NPR 100, after split it’s NPR 50, but the graph has a hole at the split time.

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In the Real World

1. eSewa/Khalti Transactions:

   • Piecewise functions model fees:

     flowchart TD
       A["Amount ≤ NPR 10,000"] -->|"0.5%"| B["Fee = 0.005 × Amount"]
       A --> C["Amount > 10,000"] -->|"1%"| D["Fee = 100 + 0.01 × (Amount - 10,000)"]

   • Discontinuity at NPR 10,000: Check if LHL = RHL.

2. Pathao/Daraz Delivery Times:

   • Step functions for distance brackets:
     • $f(d)=10$ min if $d\le 1$ km,
     • $f(d)=10+2(d-1)$ min if $d>1$ km.
   • Jump at $d=1$ km: Analyze continuity.

3. Bank Loan Interest (NMB/Global IME):

   • Piecewise interest rates:
     • $r(t)=8\%$ if loan ≤ NPR 500,000,
     • $r(t)=10\%$ if loan > 500,000.
   • Discontinuity at NPR 500,000: Critical for borrowers.

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Exam Tip

1. Graphs are your friend: Always sketch the function to visualize limits/continuity.
2. Check three conditions for continuity: $f(a)$, LHL, RHL.
3. Piecewise functions: Evaluate LHL and RHL separately at break points.
4. Common mistakes:
   • Forgetting to check if $f(a)$ is defined.
   • Assuming limits exist without verifying LHL = RHL.
5. Real-world questions: Expect applications like:
   • "Is the delivery fee function continuous?" (Pathao),
   • "Where is the stock price graph discontinuous?" (NEPSE).

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Key Formula Summary: