Exercise 12.3 | Applications of Transcendental Functions in Limits and Continuity | First Year Math 2025

Exercise 12.3 | Applications of Transcendental Functions in Limits and Continuity | First Year Math 2025 | Sir Khawar Mehmood

Mathematics is not just a subject of numbers and symbols — it’s a universal language that explains how our world works. In this lecture, Sir Khawar Mehmood brings you a complete and crystal-clear explanation of Exercise 12.3 from the New First Year Math Syllabus 2025, focusing on applications of transcendental functions to solve problems related to limits and continuity in real-world situations.

This lecture goes beyond routine calculations — it builds your conceptual foundation and helps you understand how mathematical principles are applied in growth and decay models, financial systems, and even in astronomical studies.

Exercise 12.3 | Applications of Transcendental Functions in Limits and Continuity | First Year Math 2025 | Sir Khawar Mehmood

In this detailed video lecture, Sir Khawar introduces the concept of transcendental functions, a special class of mathematical functions that include exponential, logarithmic, and trigonometric functions. These functions are called “transcendental” because they go beyond algebraic functions — they can’t be expressed as finite combinations of powers and roots.

Understanding transcendental functions is crucial because they appear everywhere in science and engineering. Whether it’s calculating population growth, predicting radioactive decay, analyzing financial investments, or modeling planetary motion — transcendental functions are the mathematical backbone behind these phenomena.

This lecture connects theory to real-life applications so that students not only learn how to solve questions in the exam but also appreciate how mathematics shapes our world.

📘 Topics Covered in Exercise 12.3 (New Syllabus 2025)

1. Introduction to Transcendental Functions

What are transcendental functions?

Difference between algebraic and transcendental functions

Examples: exponential, logarithmic, and trigonometric functions

Real-world significance of transcendental behavior

2. Application of Transcendental Functions in Limits and Continuity

How limits are used to understand function behavior near specific points

Continuity and differentiability of transcendental functions

Techniques for evaluating limits involving exponential and logarithmic terms

Visual understanding using function graphs

3. Growth and Decay Problems

Modeling natural growth (like population increase)

Radioactive decay and half-life calculations

Newton’s Law of Cooling and natural process modeling

Using exponential models to predict real-life outcomes

4. Applications in Finance and Economics

Compound interest formula derived from exponential growth

Continuous compounding and its connection to the constant “e”

Predicting future values of investments using logarithmic and exponential equations

Real examples from banking and business economics

5. Applications in Astronomy and Physics

Modeling intensity of light and sound using exponential decay

Application of transcendental functions in planetary motion and orbital mechanics

Understanding inverse-square laws using continuous mathematical models

The importance of limits and continuity in celestial calculations

PDF SOLUTION:

6. Complete Step-by-Step Solutions of Exercise 12.3

Each question explained with clear reasoning

Logical steps, simplified calculations, and graphical interpretation

Concept building through real examples and applied mathematics

🧠 Understanding Transcendental Functions

Transcendental functions are not just formulas; they describe how natural systems behave. For instance, the exponential function (e^x) represents continuous growth or decay. When we apply limits to such functions, we understand their long-term or instantaneous behavior — whether something is increasing, approaching zero, or stabilizing.

In Exercise 12.3, you’ll see how limits help in predicting outcomes when variables approach infinity or zero. This helps in understanding population saturation, chemical decay, or investment stability. Continuity ensures that the model behaves smoothly without sudden jumps — exactly how real processes behave in nature.

Through the lecture, Sir Khawar uses easy explanations, visual examples, and logical reasoning to make complex mathematical behavior simple and interesting.

📈 Growth and Decay – Real-World Mathematical Models

Growth and decay models are fundamental applications of transcendental functions. They describe systems that increase or decrease proportionally to their current state. For example:

Population Growth:

where is the initial population and is the growth constant.

This model helps predict future population based on current trends.

Radioactive Decay:

where is the decay constant.

This equation is widely used in physics, chemistry, and even medicine to measure how substances transform over time.

Sir Khawar not only teaches how to solve these problems mathematically but also connects each concept with how it works in real life — making learning both practical and engaging.

💰 Applications in Finance and Economics

In finance, exponential and logarithmic functions are vital for understanding growth over time. From compound interest to inflation and investment modeling, transcendental functions form the mathematical foundation of economics.

Compound Interest Formula and for continuous compounding:

Here, the mathematical constant e (≈ 2.718) comes from the natural exponential function — a key transcendental function.

Sir Khawar explains how these formulas are derived using limits and how the concept of continuity ensures that money grows smoothly over time rather than in sudden jumps.

By understanding this, students learn how mathematics is used in banking, economics, and business planning — giving practical meaning to abstract formulas.

🌌 Applications in Astronomy and Space Science

The role of transcendental functions extends to the universe itself. In astronomy, they describe how light intensity diminishes with distance, how planets move around stars, and how gravitational forces vary continuously across space.

For example:

Intensity of Light (Inverse Square Law):

shows how light intensity decays exponentially as distance increases.

Planetary Motion:

Trigonometric transcendental functions model the circular and elliptical orbits of celestial bodies.

These concepts help scientists predict planetary behavior, spacecraft trajectory, and the lifespan of stars — all using the same mathematical principles you learn in Exercise 12.3.

Sir Khawar’s approach makes such vast scientific ideas understandable by connecting them directly to the mathematical functions in your textbook.

🔹 Importance of Limits and Continuity

Limits and continuity are not just abstract mathematical ideas — they describe smooth, predictable change in any system. In real life, physical, biological, and economic systems rarely jump abruptly from one state to another; they evolve gradually.

This gradual change is mathematically expressed through continuous functions and analyzed using limits.

Through examples in Exercise 12.3, you’ll learn:

How to calculate limits involving exponential and logarithmic functions

How to test the continuity of transcendental functions

Why discontinuities cause physical or practical problems in modeling

By mastering these ideas, you not only improve your exam performance but also develop an intuitive understanding of how mathematical continuity reflects real-world behavior.

🎓 Why This Lecture is Important

This lecture by Sir Khawar Mehmood is not just another math video — it’s a bridge between textbook theory and real-life application. It helps students:

Strengthen their grasp of transcendental functions

Understand how to apply limits and continuity in real-world problems

Prepare effectively for board exams 2025

Develop analytical skills for higher-level studies in physics, engineering, and economics

Every question in Exercise 12.3 is explained step-by-step, ensuring students can confidently solve similar problems in exams and competitive tests.

🧩 What You’ll Gain from This Lecture

✅ Conceptual clarity of transcendental functions

✅ Practical understanding of growth and decay models

✅ Knowledge of real-life financial and astronomical applications

✅ Complete solutions of Exercise 12.3 (First Year Math 2025)

✅ Tips for board exam preparation and problem-solving

✅ Connection between mathematical theory and practical world

📺 Watch Now on Math’s Universe Online

This comprehensive lecture is available on Math’s Universe Online YouTube Channel, where Sir Khawar Mehmood continues his mission to make mathematics easy, logical, and inspiring for every student.

Join thousands of students who are learning mathematics with understanding — not just memorization.

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Exercise 12.3 | Applications of Transcendental Functions in Limits and Continuity | First Year Math 2025 | Sir Khawar Mehmood