Exercise 12.1 | New First Year Math 2025 | Limit of a Function | Complete Solution and Explanation

Introduction:

In the new 11th class Mathematics syllabus 2025, Chapter 12 is about Limit and Continuity. The very first exercise, Exercise 12.1, introduces the basic and most important concept of calculus: the limit of a function. Before learning about continuity or derivatives, students must first understand what a limit means. This exercise explains the idea of a limit in simple language, introduces standard notations, gives numerical and graphical approaches, and provides the fundamental theorems that make solving problems easier.

Understanding limits is important because many concepts in higher mathematics, such as differentiation, integration, and analysis, are based on it. For example, when we want to know the slope of a curve at a single point, or the behavior of a function near infinity, the concept of a limit is used.

Meaning of the Phrase “x approaches a”

The phrase “x approaches a” means that the value of x gets closer and closer to a certain number a, but does not necessarily become equal to it. For example, when we say “x approaches 0”, it means x is moving nearer and nearer to 0, but not exactly equal to 0.

Similarly:

x approaches 0 → x is very close to zero.

x approaches infinity → x increases without any bound.

x approaches a → x moves closer to a particular number a.

Limit of a Function

Suppose f(x) is a function. We say the limit of f(x) as x approaches a is equal to L if the values of f(x) get closer and closer to L as x gets closer and closer to a. Symbolically, this is written as:

limit of f(x) as x approaches a = L

In simple words: the limit tells us the value that a function “wants to be” as x comes close to some number.

Numerical Approach to Limits

Sometimes we cannot directly put the value of x into a function because it may make the denominator zero or create an undefined form. In such cases, we use a numerical approach. That means we put values of x closer and closer to the required number and check the result.

For example, if we want to find the limit of (x^2 - 1)/(x - 1) as x approaches 1, we cannot directly substitute x = 1 because the denominator becomes zero. Instead, we take values near 1 like 0.9, 0.99, 1.1, 1.01 and see what happens. In this case, the function values get closer and closer to 2. So, the limit is 2.

Theorems on Limits of Functions

These theorems are very helpful because they make limit questions easy to solve. Instead of checking every time with a table of values, we can use these rules.

1. Limit of a constant: limit of c as x approaches a = c

2. Limit of identity function: limit of x as x approaches a = a

3. Limit of a power function: limit of x^n as x approaches a = a^n

4. Constant multiple rule: limit of c f(x) = c * limit of f(x)

5. Sum and difference rule: limit of [f(x) ± g(x)] = limit of f(x) ± limit of g(x)

6. Product rule: limit of [f(x) * g(x)] = (limit of f(x)) * (limit of g(x))

7. Quotient rule: limit of f(x)/g(x) = (limit of f(x)) / (limit of g(x)), provided the denominator is not zero

8. Composition rule: limit of f(g(x)) = f(limit of g(x))

9. Limit of a polynomial function: directly substitute the value of x into the polynomial

10. Limit of a rational function: substitute the value of x into numerator and denominator, if the denominator is not zero

These theorems cover almost every type of question that comes in Exercise 12.1.

Important Results of Limits:

There are some important standard limits that students must remember because they are frequently used:

1. limit of (sin x / x) as x approaches 0 = 1

2. limit of (1 + 1/x)^x as x approaches infinity = e

3. limit of (a^x - 1)/x as x approaches 0 = log a

4. limit of (1 - cos x)/x^2 as x approaches 0 = 1/2

5. limit of (tan x / x) as x approaches 0 = 1

These results are very useful in solving not only Exercise 12.1 but also later chapters on derivatives.

Limit at Infinity

Sometimes we are interested in what happens to a function as x becomes very large. This is called the limit at infinity.

Examples:

limit of 1/x as x approaches infinity = 0

limit of (x^2 + 1)/x^2 as x approaches infinity = 1

limit of (2x + 3)/(x + 1) as x approaches infinity = 2

Limit of a Sequence

A sequence is like a list of numbers that follow a rule. If the terms of a sequence get closer and closer to some fixed number, that number is called the limit of the sequence.

For example:

The sequence 1, 1/2, 1/3, 1/4, … has limit 0.

The sequence (1 + 1/n)^n has limit e.

Exercise 12.1 Complete Solutions 

In this exercise, students practice applying the rules of limits, important results, and direct substitution. The main methods used are:

1. Direct substitution into the function (when denominator is not zero).

2. Simplification by factorization (for example, canceling (x - a) terms).

3. Applying the theorems of limits (sum, product, quotient, etc.).

4. Using important results like sin x / x = 1.

5. Considering behavior at infinity.

6. Numerical approach when necessary.

Every question in this exercise is based on these ideas. By carefully applying these steps, all limits can be solved easily.

Conclusion

The study of limits is the foundation of calculus. In this chapter, students learn what it means when we say “x approaches a number”, how to evaluate the limit of a function, and why it is useful in mathematics. Theorems on limits and important standard results make it possible to solve questions in a short and simple way. Exercise 12.1 provides the first practical training in applying these techniques. Once a student masters this exercise, they will find later topics like continuity, derivatives, and integrals much easier.

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