1st Year Math Chapter 12 Exercise 12.3 Notes: Applications of Limits (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 12.3 closes Chapter 12, Limit and Continuity, by applying the limit and continuity concepts from Exercises 12.1 and 12.2 to real-world situations — modeling growth, decay, and cost using limits at infinity, and checking whether practical piecewise functions are continuous.
What Does This Exercise Cover?
This exercise shows that limits aren’t just an abstract algebra exercise — they describe genuinely useful things: what a quantity settles toward in the long run, and whether a real-world rule (like a pricing structure) behaves predictably at the point where it changes.
Key Concepts
1. Limits at Infinity
A limit at infinity, written lim(x→∞) f(x), describes what value a function approaches as its input grows without bound. In applications, this describes the long-term or ‘settling’ behavior of a quantity over time or distance.
2. Exponential Decay Models
A quantity that decays exponentially is modeled as A(t) = A₀e^(−kt), where A₀ is the initial amount and k > 0 is the decay rate. Since e^(−kt) → 0 as t → ∞, the limit of A(t) as t → ∞ is always 0 — the quantity shrinks toward (but never quite reaches) zero.
3. Logistic (Bounded) Growth Models
Many real populations don’t grow forever — they level off at a maximum sustainable value called the carrying capacity. This is modeled as P(t) = L / (1 + Ae^(−kt)), where L is the carrying capacity. As t → ∞, the exponential term vanishes, and P(t) → L.
4. Limits of Rational Functions at Infinity
For a rational function like (at)/(t + b), the limit as t → ∞ is found by dividing the numerator and denominator by the highest power of t present. This technique reveals the value a business quantity (like profit or output) approaches as time or scale increases indefinitely.
5. Inverse-Power Models
Quantities that weaken with distance, like signal strength or light intensity, are often modeled as k/dⁿ. Since the denominator grows without bound as d → ∞, the limit of such a model is always 0 — the effect fades to nothing at great distance.
6. Continuity in Practical Piecewise Functions
Real-world cost or pricing rules are often defined piecewise (a different formula above and below some threshold). The same continuity test from Exercise 12.2 — checking that the left-hand limit, right-hand limit, and function value all agree — reveals whether such a rule changes smoothly at that threshold, or jumps unexpectedly.
Solved Examples
Example 1: A radioactive sample has mass M(t) = 200e^(−0.2t) grams, t in years. Find lim(t→∞) M(t) and interpret the result.
As t → ∞, −0.2t → −∞, so e^(−0.2t) → 0.
Therefore M(t) = 200 × e^(−0.2t) → 200 × 0.
Answer: lim(t→∞) M(t) = 0 — the sample’s mass decays toward zero over time
Example 2: A bacteria colony grows according to N(t) = 5000 / (1 + 4e^(−0.3t)). Find the long-term population.
As t → ∞, e^(−0.3t) → 0.
The denominator approaches 1 + 4(0) = 1.
So N(t) → 5000 / 1.
Answer: The population levels off at a carrying capacity of 5000
Example 3: A company’s monthly profit (in thousands of rupees) is modeled by P(t) = 800t / (t + 20). Find lim(t→∞) P(t) and interpret it.
Divide numerator and denominator by t: P(t) = 800 / (1 + 20/t).
As t → ∞, 20/t → 0, so the denominator approaches 1.
Answer: lim(t→∞) P(t) = 800 — monthly profit approaches a ceiling of Rs. 800,000
Example 4: The intensity of light from a source is I(d) = 2000/d² lux. Find I(5) and lim(d→∞) I(d).
I(5) = 2000/5² = 2000/25.
As d → ∞, d² → ∞, so 2000/d² → 0.
Answer: I(5) = 80 lux; lim(d→∞) I(d) = 0 — the light fades to nothing at great distance
Example 5: A shipping company charges C(x) = 15x + 200 for x ≤ 20 kg, and C(x) = 15x + 250 for x > 20 kg. Determine whether C is continuous at x = 20.
C(20) = 15(20) + 200 = 300 + 200 = 500.
LHL: lim(x→20⁻) (15x + 200) = 300 + 200 = 500.
RHL: lim(x→20⁺) (15x + 250) = 300 + 250 = 550.
Since LHL (500) ≠ RHL (550), the two one-sided limits disagree.
Answer: C is discontinuous at x = 20 — a customer’s cost jumps by Rs. 50 the instant weight crosses 20 kg
Sample MCQs
1. As t → ∞, e^(−kt) (for k > 0) approaches:
a) ∞ b) 0 c) 1 d) k
Answer: b) 0
2. In a logistic growth model P(t) = L / (1 + Ae^(−kt)), the value L represents:
a) The initial population b) The long-term carrying capacity c) The growth rate d) The decay constant
Answer: b) The long-term carrying capacity
3. For a rational function like at/(t + b) as t → ∞, the limit is found by:
a) Substituting t = 0 b) Dividing numerator and denominator by the highest power of t c) Multiplying by t d) Taking the derivative
Answer: b) Dividing numerator and denominator by the highest power of t
4. An inverse-power model k/dⁿ (n > 0) approaches, as d → ∞:
a) ∞ b) k c) 0 d) 1
Answer: c) 0
5. For a piecewise cost function to be continuous at a price-break point, its left-hand limit, right-hand limit, and function value must:
a) All be infinite b) All be equal to each other c) All be zero d) All be different
Answer: b) All be equal to each other
Important Short Questions
- What does lim(t→∞) f(t) describe about a real-world model?
- In an exponential decay model A(t) = A₀e^(−kt), what does the limit as t → ∞ represent physically?
- Explain what the ‘carrying capacity’ represents in a logistic growth model.
- Find lim(t→∞) of (300t)/(t + 15), showing your method.
- Explain why a business might want to check the continuity of a piecewise cost function at a price-break point.
Important Long Questions
- A radioactive sample decays according to M(t) = 200e^(−0.2t). Find lim(t→∞) M(t) and explain what it means physically.
- A bacteria population follows N(t) = 5000 / (1 + 4e^(−0.3t)). Find the long-term population and describe the general shape of this growth model.
- A company’s profit is modeled by P(t) = 800t/(t + 20) (in thousands of rupees). Find lim(t→∞) P(t) and interpret the result in business terms.
- A shipping company charges C(x) = 15x + 200 for x ≤ 20 kg and C(x) = 15x + 250 for x > 20 kg. Determine whether this cost function is continuous at x = 20, and explain what a discontinuity like this means for a customer whose package is right at the boundary.
How to Approach This Exercise Effectively
- Identify what kind of model you’re looking at first (decay, bounded growth, rational, or inverse-power) — each has a predictable limiting behavior as the variable grows large.
- For exponential terms like e^(−kt), remember the sign of the exponent tells you everything: negative exponents shrink to 0, positive exponents grow without bound.
- For rational functions, always divide through by the highest power in the denominator before taking the limit — guessing the answer by inspection is unreliable for anything but the simplest cases.
- When a question asks you to interpret a limit, always translate the number back into the real-world quantity it represents (a ceiling, a floor, a fade-out), not just report the number itself.
- For continuity questions on piecewise real-world functions, apply the exact three-condition test from Exercise 12.2 — don’t treat it as a new or different method.
FAQs
Q: Why do real-world models often use limits at infinity?
A: Because many practical questions are really about long-term behavior — where a population settles, what a cost eventually approaches, or how weak a signal becomes at distance — and a limit at infinity captures exactly that.
Q: What’s the difference between exponential decay and logistic growth in terms of their long-term limit?
A: Exponential decay always settles toward zero, while logistic growth settles toward a fixed positive carrying capacity — the two models describe fundamentally different long-term outcomes.
Q: Why might a business’s cost function be continuous, or accidentally discontinuous, at a price-break point?
A: A well-designed pricing structure is usually built to be continuous so costs change smoothly; a discontinuity often reveals a pricing rule that wasn’t carefully matched at the boundary, creating an abrupt jump right at the threshold.
Q: Can a quantity modeled by a limit at infinity ever actually reach that limiting value?
A: In the strict mathematical model, usually not — the quantity gets arbitrarily close without ever exactly equaling the limit, though in practice the difference becomes too small to matter after enough time or distance.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 12: Limit and Continuity, Exercise 12.3.