Physics and Mathematics
Limits
1. Concept Overview
In mathematics, especially in calculus, the idea of a limit helps us understand the behavior of a function near a point, even if the function is not defined at that point.
In simple words:
A limit tells us what value a function approaches as the variable gets closer and closer to a certain number.
Limits form the foundation of differentiation and integration, making them one of the most important concepts in Class 12 Mathematics.
2. Meaning of a Limit (Intuitive Explanation)
Consider a function [f(x)].
We study what happens to [f(x)] when [x] gets very close to a number [a].
- We do not necessarily care about the value at [x = a]
- We care about the values of [f(x)] near [a]
Mathematically, this is written as:
[\lim_{x \to a} f(x)]
This reads as:
“The limit of [f(x)] as [x] approaches [a]”
3. Understanding “Approaches” (Very Important)
The word approaches does not mean equals.
- [x \to a] means:
- comes very close to [a]
- but [x ≠ a]
Example:
If [x \to 2], then [x] can be:
- 1.9, 1.99, 1.999 (from left)
- 2.1, 2.01, 2.001 (from right)
4. Formal Definition
If the values of [f(x)] become closer and closer to a real number [L] as [x] approaches [a], then:
[\lim_{x \to a} f(x) = L]
This means:
- The function is approaching [L]
- Even if [f(a)] is:
- undefined, or
- different from [L]
5. Approaching from Left and Right
A number can be approached in two different ways:
- From the left side (values less than [a])
- From the right side (values greater than [a])
This leads to the concepts of Left-Hand Limit and Right-Hand Limit.
6. Left-Hand Limit (LHL)
The Left-Hand Limit of a function at [x = a] is the value that the function approaches when [x] approaches [a] from values less than [a].
Mathematically:
[\lim_{x \to a^-} f(x)]
Here:
- [x \to a^-] means [x < a]
- approaches [a] from the left side
Example (LHL)
If [f(x)] is observed for values like:
- [a − 0.1], [a − 0.01], [a − 0.001]
Then we are finding the Left-Hand Limit.
7. Right-Hand Limit (RHL)
The Right-Hand Limit of a function at [x = a] is the value that the function approaches when [x] approaches [a] from values greater than [a].
Mathematically:
[\lim_{x \to a^+} f(x)]
Here:
- [x \to a^+] means [x > a]
- approaches [a] from the right side
Example (RHL)
If [f(x)] is observed for values like:
- [a + 0.1], [a + 0.01], [a + 0.001]
Then we are finding the Right-Hand Limit.
8. Existence of a Limit (Very Important Rule)
A limit of a function at [x = a] exists if and only if:
[\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)]
If both LHL and RHL exist and are equal, then:
[\lim_{x \to a} f(x)] exists.
If they are not equal, then the limit does not exist.
9. Limit vs Function Value
This is a very common source of confusion.
| Concept | Meaning |
|---|---|
| LHL | Behaviour from left |
| RHL | Behaviour from right |
| Limit | Common value of LHL and RHL |
| Function value | Exact value at the point |
Important:
A function value may exist even when the limit does not exist.
8. Graphical Interpretation (Conceptual)
- If the graph approaches the same height from left and right → limit exists
- If the graph approaches different heights → limit does not exist
- If there is a hole but same approach → limit exists
9. Key Features of Limits, LHL, and RHL
- Limits study approaching behavior, not exact substitution
- LHL and RHL must be equal for the limit to exist
- Function need not be defined at the point
- Essential for understanding continuity, differentiation, and integration
6. Types of Limits (Introductory)
At this stage, we broadly talk about:
- Finite limits
- Infinite limits
- Limits leading to indeterminate forms
(Each will be studied in detail later.)
7. Conceptual Questions with Solutions
1. What is meant by the limit of a function?
The limit of a function describes the value that the function approaches when the input variable gets closer to a given number. It focuses on nearby values, not the value at the point itself.
2. Does [x \to a] mean [x = a]?
No. It means [x] gets very close to [a] but never becomes exactly equal to [a].
3. Can a limit exist if the function is not defined at that point?
Yes. The limit depends on values near the point, not on the value at the point.
4. Can a function have a value but no limit?
Yes. If left-hand and right-hand limits are different, the limit does not exist even though the function value exists.
5. Why are limits important in calculus?
Limits form the foundation of differentiation, integration, and continuity.
6. What does [\lim_{x \to a} f(x) = L] indicate?
It indicates that [f(x)] approaches [L] as [x] gets closer to [a].
7. Is limit a process or a value?
It is a process of approaching that leads to a value.
8. Can limits be negative?
Yes. Limits can be positive, negative, or zero.
9. What role do limits play in handling division by zero?
Limits allow us to study behavior near zero without actual division by zero.
10. Are limits only related to algebra?
No. Limits are used in algebra, trigonometry, calculus, and physics.
11. Can limits be applied to real-life situations?
Yes, especially in motion, growth, and change-related problems.
12. Is the limit always equal to function value?
No. They may or may not be equal.
13. What happens if function values oscillate near a point?
The limit may not exist.
14. Can limits be infinite?
Yes. If function values grow without bound.
15. Are limits studied before continuity for a reason?
Yes. Continuity is defined using limits.
8. FAQ / Common Misconceptions
1. Limit means substitution.
Incorrect. Limits are about approaching, not substituting.
2. Limit and function value are always same.
False. They can be different.
3. If a function is undefined, limit cannot exist.
Wrong. Limits may exist without function value.
4. Limits are only theoretical.
No. They are widely used in science and engineering.
5. Limit exists only when graph is continuous.
Incorrect.
6. Approaching means equal.
Wrong interpretation.
7. Limits are difficult and abstract.
They are intuitive when understood conceptually.
8. Infinite limit means limit does not exist.
Not true. It is a valid type of limit.
9. Limit applies only at numbers.
Limits can be studied at infinity as well.
10. Limits are optional for calculus.
Limits are essential.
9. Practice Questions
(Identifying Limits – No Evaluation)
Question 1. Write the meaning of [\lim_{x \to 3} f(x)] in words.
Step-by-Step Solution:
[x \to 3] means [x] approaches 3.
[f(x)] represents function values near 3.
We observe behavior near 3, not at 3.
Conclusion:
It represents the value approached by [f(x)] as [x] comes close to 3.
Question 2. Does the function value at [x = a] affect the limit?
Step-by-Step Solution:
Limits depend on nearby values.
Function value is not compulsory.
Conclusion:
Function value does not affect the existence of a limit.
Question 3. Can a limit exist even if [f(a)] is undefined?
Step-by-Step Solution:
Limits consider values near [a].
Definition at [a] is not required.
Conclusion:
Yes, a limit can exist.
Question 1. Identify LHL and RHL (exist or not) for:
[\lim_{x \to 0} |x|]
Step-by-Step Solution:
Consider values of [x] approaching 0 from the left side ([x < 0]). For negative values of [x], [|x| = −x]. As [x \to 0^-], the value of [|x|] approaches [0]. Now consider values of [x] approaching 0 from the right side ([x > 0]).
For positive values of [x], [|x| = x].
As [x \to 0^+], the value of [|x|] also approaches [0].
Conclusion:
Both LHL and RHL exist and are equal. Hence, the limit exists.
Question 2. Identify LHL and RHL for:
[\lim_{x \to 0} \dfrac{|x|}{x}]
Step-by-Step Solution:
For LHL ([x \to 0^-]):
When [x < 0], [|x| = −x]. So, [\dfrac{|x|}{x} = \dfrac{-x}{x} = -1]. Thus, as [x \to 0^-], the function approaches [-1]. For RHL ([x \to 0^+]): When [x > 0], [|x| = x].
So, [\dfrac{|x|}{x} = 1].
Thus, as [x \to 0^+], the function approaches [1].
Conclusion:
LHL and RHL both exist but are not equal.
Therefore, the limit does not exist.
Question 3. Identify LHL and RHL for:
[\lim_{x \to 1} \dfrac{x – 1}{|x – 1|}]
Step-by-Step Solution:
For LHL ([x \to 1^-]):
[x − 1 < 0], so [|x − 1| = −(x − 1)]. Hence, the expression becomes [-1]. For RHL ([x \to 1^+]): [x − 1 > 0], so [|x − 1| = x − 1].
Hence, the expression becomes [1].
Conclusion:
LHL and RHL are different, so the limit does not exist.
Question 4. Identify LHL and RHL for:
[\lim_{x \to 0} \dfrac{1}{x}]
Step-by-Step Solution:
For LHL ([x \to 0^-]):
[x] is negative and very small.
So, [\dfrac{1}{x}] becomes a large negative value.
For RHL ([x \to 0^+]):
[x] is positive and very small.
So, [\dfrac{1}{x}] becomes a large positive value.
Conclusion:
LHL and RHL both tend to infinity but in different directions.
Hence, the limit does not exist.
Question 5. Identify LHL and RHL for:
[\lim_{x \to 0} x \sin \dfrac{1}{x}]
Step-by-Step Solution:
As [x \to 0^-], [x] approaches 0 and [\sin(1/x)] oscillates between [-1] and [1].
Their product approaches [0] from the left side.
As [x \to 0^+], [x] approaches 0 and [\sin(1/x)] still oscillates between [-1] and [1].
Their product again approaches [0] from the right side.
Conclusion:
Both LHL and RHL exist and are equal.
Hence, the limit exists.
Question 6. Identify LHL and RHL for:
[\lim_{x \to 2} \lfloor x \rfloor]
Step-by-Step Solution:
For LHL ([x \to 2^-]):
Values slightly less than 2 have greatest integer value [1].
For RHL ([x \to 2^+]):
Values slightly greater than 2 have greatest integer value [2].
Conclusion:
LHL and RHL exist but are not equal.
Hence, the limit does not exist.
Question 7. Identify LHL and RHL for:
[\lim_{x \to 0} \sqrt{x^2}]
Step-by-Step Solution:
For any value of [x], [\sqrt{x^2} = |x|].
As [x \to 0^-], [|x| \to 0].
As [x \to 0^+], [|x| \to 0].
Conclusion:
Both LHL and RHL exist and are equal.
Therefore, the limit exists.
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