Limits, Continuity, and Differentiability: Core Concepts & Applications

1. Limits: The Heart of Calculus

1.1 Definition and Intuition

A limit describes the value a function $f(x)$ approaches as $x$ approaches a point $a$ (or infinity). It is written as: $$\underset{x\to a}{\lim }f(x)=L$$ Key Idea: The limit exists if the left-hand limit ($x\to a^{-}$) and right-hand limit ($x\to a^{+}$) are equal and finite.

Worked Example 1: Evaluate ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$. Solution:

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

Visual: The graph has a hole at $x=2$, but the limit is the $y$-value the function approaches (4).

1.2 Special Limits and Techniques

Worked Example 2: Evaluate ${\lim }_{x\to \infty }\frac{3x^2+2x-1}{2x^2-5}$. Solution:

1. Divide numerator/denominator by $x^2$: $$\frac{3+\frac{2}{x}-\frac{1}{x^2}}{2-\frac{5}{x^2}}$$
2. As $x\to \infty$, terms with $x$ in the denominator vanish: $$\frac{3+0-0}{2-0}=\frac{3}{2}$$

Real-World Tie: Pathao’s Ride Pricing Pathao’s surge pricing algorithm uses limits to model demand spikes. If demand $D(x)$ grows quadratically with time $x$, the limit ${\lim }_{x\to \infty }\frac{D(x)}{x}$ helps predict long-term pricing adjustments.

2. Continuity: Smoothness of Functions

2.1 Definition

A function $f(x)$ 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:

graph TD
    A["Discontinuity Types"] --> B["Removable (Hole)"]
    A --> C["Jump (Left ≠ Right)"]
    A --> D["Infinite (Vertical Asymptote)"]
    A --> E["Essential (Wild Behavior)"]

Worked Example 3: Check continuity of $f(x)=\{\begin{array}{ll}\frac{\sin x}{x}& x\ne 0\\ 1& x=0\end{array}$ at $x=0$. Solution:

1. $f(0)=1$ (defined).
2. ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ (standard limit).
3. Since ${\lim }_{x\to 0}f(x)=f(0)$, $f(x)$ is continuous at $x=0$.

2.2 Intermediate Value Theorem (IVT)

Statement: If $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c\in (a,b)$ such that $f(c)=N$.

Application: NTC’s Network Coverage Optimization NTC uses IVT to ensure signal strength $S(x)$ (continuous in terrain $x$) meets a minimum threshold $S_{\text{min}}$ between two towers. If $S(a)\ge S_{\text{min}}$ and $S(b)\le S_{\text{min}}$, IVT guarantees a point $c$ where $S(c)=S_{\text{min}}$.

3. Differentiability: The Derivative’s Domain

3.1 Definition

A function $f(x)$ is differentiable at $x=a$ if the derivative $f^{\prime }(a)$ exists, i.e.: $$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$$ Key Insight: Differentiability implies continuity, but not vice versa.

Worked Example 4: Show $f(x)=|x|$ is not differentiable at $x=0$. Solution:

1. Left derivative: $f_{-}^{\prime }(0)={\lim }_{h\to 0^{-}}\frac{|0+h|-|0|}{h}=-1$.
2. Right derivative: $f_{+}^{\prime }(0)={\lim }_{h\to 0^{+}}\frac{|0+h|-|0|}{h}=1$.
3. Since $f_{-}^{\prime }(0)\ne f_{+}^{\prime }(0)$, $f(x)$ is not differentiable at $x=0$.

Real-World Tie: Khalti’s Transaction Fees Khalti’s fee function $F(x)$ (where $x$ = transaction amount) is continuous but not differentiable at $x=1000$ (a threshold for fee jumps). This creates a "corner" in the graph, reflecting abrupt policy changes.

4. Relationships Between Concepts

graph LR
    A["Differentiable at a"] --> B["Continuous at a"]
    B --> C["Limit exists at a"]
    C --> D["f(a) is defined"]
    D --> E["No guarantee of differentiability"]

Comparison Table:

In the Real World

1. eSewa’s Bill Payments

   • Limit Concept: When you pay a bill, eSewa’s backend calculates the limit of transaction fees as the amount $x\to \infty$. For example: $$\underset{x\to \infty }{\lim }\frac{\text{Fee}(x)}{x}=0.01$$ ensures fees don’t grow faster than 1% of the amount.

2. Daraz’s Order Fulfillment Queue

   • Continuity: Daraz’s warehouse inventory $I(t)$ is modeled as a continuous function. If $I(8\text{AM})=500$ units and $I(10\text{AM})=300$ units, the IVT guarantees a time $t$ when $I(t)=400$ units (useful for restock alerts).

3. NEPSE Stock Prices

   • Differentiability: Stock prices $P(t)$ are continuous but not differentiable at market open/close times (sharp jumps). Traders use this to identify volatility spikes.

Exam Tip

1. Limit Evaluation:

   • Always check direct substitution first. If it fails, try factoring, rationalizing, or L'Hôpital’s Rule.
   • For $\infty$ limits, dominate by the highest power of $x$.

2. Continuity Proofs:

   • Verify three conditions: $f(a)$ exists, limit exists, and they are equal.
   • IVT questions: Draw a number line and mark $f(a)$ and $f(b)$. If $N$ is between them, conclude existence.

3. Differentiability:

   • Corner points (like $|x|$) and cusps (like $x^{2/3}$) are common traps. Check left and right derivatives.
   • Graphical questions: Sketch the function to spot discontinuities or non-differentiable points.

4. Common Pitfalls:

   • Assuming continuity implies differentiability (e.g., $f(x)=|x|$ at $x=0$).
   • Ignoring one-sided limits (e.g., ${\lim }_{x\to 0^{-}}\frac{1}{x}\ne {\lim }_{x\to 0^{+}}\frac{1}{x}$).
   • Misapplying L'Hôpital’s Rule to non-indeterminate forms (e.g., $\frac{0}{1}$).

Final Note: Master these concepts by:

• Sketching graphs for every function (identify holes, jumps, asymptotes).
• Practicing limit laws (sum, product, quotient rules).
• Connecting to real systems (e.g., how banks use limits for loan amortization).