Physics and Mathematics
Indeterminate Form
1. Concept Overview
In the study of Limits, an indeterminate form arises when direct substitution of the limiting value into a function does not give a meaningful result. Such forms do not determine the limit uniquely and require further simplification or analytical methods.
Commonly encountered indeterminate forms are:
- [0/0]
- [∞/∞]
- [0·∞]
- [∞ − ∞]
- [0^0]
- [1^∞]
- [∞^0]
These forms indicate that the limit cannot be evaluated directly and must be resolved using algebraic manipulation or limit laws.
2. Explanation and Mathematical Understanding
When evaluating a limit [\lim_{x \to a} f(x)], we usually try direct substitution. Three situations may arise:
- The limit exists and is finite → directly substitutable.
- The limit is infinite → not indeterminate.
- The result is ambiguous (indeterminate form) → needs simplification.
Example 1: [0/0] Form
Consider:
[\lim_{x \to 2} \dfrac{x^2 − 4}{x − 2}]
Direct substitution gives:
Numerator = [0], Denominator = [0] → form [0/0]
This does not mean the limit is zero. It means the limit is undefined at this stage and needs simplification.
Factorizing:
[x^2 − 4 = (x − 2)(x + 2)]
So,
[\lim_{x \to 2} \dfrac{(x − 2)(x + 2)}{x − 2} = \lim_{x \to 2} (x + 2) = 4]
Hence, the indeterminate form is resolved.
3. Dimensions and Units
Limits are purely mathematical concepts.
- They do not have physical dimensions or units.
- If applied to a physical quantity, the resulting limit carries the same unit as the function.
4. Key Features of Indeterminate Forms
- Indeterminate forms do not give final answers.
- They indicate the need for simplification.
- Different simplifications may lead to different limit values.
- Appearance of an indeterminate form does not guarantee that the limit does not exist.
- Most limit problems in Class 12 calculus start by identifying an indeterminate form.
5. Important Indeterminate Forms to Remember
| Expression | Indeterminate Form |
|---|---|
| [\dfrac{0}{0}] | Yes |
| [\dfrac{∞}{∞}] | Yes |
| [0·∞] | Yes |
| [∞ − ∞] | Yes |
| [0^0] | Yes |
| [1^∞] | Yes |
| [∞^0] | Yes |
6. Conceptual Questions with Solutions
1. What is meant by an indeterminate form in limits?
An indeterminate form occurs when direct substitution of the limiting value results in an expression that does not give a definite value, such as [0/0] or [∞/∞]. These forms do not determine the limit uniquely and indicate the need for further simplification.
2. Why does [0/0] not mean the limit is zero?
The form [0/0] only shows that both numerator and denominator approach zero. Depending on how fast they approach zero, the limit may be finite, infinite, or may not exist at all. Hence, it gives no definite information.
3. Can a limit exist even if it gives an indeterminate form initially?
Yes. Many limits initially give indeterminate forms but become well-defined after simplification, such as factorisation or rationalisation.
4. Is [∞] an indeterminate form?
No. [∞] represents a determinate result, indicating that the function increases without bound. Only ambiguous expressions like [∞ − ∞] or [∞/∞] are indeterminate.
5. Why is [∞ − ∞] indeterminate?
Both quantities increase without bound, but their difference may be finite, zero, or infinite depending on their rates of growth. Hence, no definite conclusion can be drawn directly.
6. Why is [∞/∞] indeterminate?
The ratio of two quantities tending to infinity depends on which grows faster. Without simplification, the limit cannot be predicted.
7. Is [0·∞] always zero?
No. Although one factor approaches zero and the other infinity, their product may be finite, zero, or infinite depending on the function.
8. Why is [0^0] considered indeterminate?
In limits, both the base and exponent approach zero. The resulting value depends on how they approach zero, so the limit is not uniquely determined.
9. Why is [1^∞] an indeterminate form?
Although [1^n = 1], in limits the base approaches 1 and the exponent approaches infinity. Small deviations from 1 can significantly affect the result.
10. Why is [∞^0] indeterminate?
An infinitely large base raised to a power approaching zero can give different results depending on their rates of change.
11. Do all indeterminate forms imply discontinuity?
No. Many functions are continuous even though their limits initially appear indeterminate.
12. Is simplification always necessary when an indeterminate form appears?
Yes. Algebraic manipulation or standard limits are essential to evaluate such limits.
13. Can two different functions give the same indeterminate form but different limits?
Yes. The final limit depends on the function itself, not merely on the indeterminate form.
14. Are indeterminate forms only studied in limits?
Primarily yes, because limits deal with approaching values rather than exact substitution.
15. Which indeterminate forms are most common in Class 12 exams?
The most common ones are [0/0] and [∞/∞], usually resolved using factorisation or dominant terms.
7. FAQ / Common Misconceptions
1. [0/0] means the answer is zero.
Incorrect. It gives no information about the limit and must always be simplified.
2. If substitution gives [∞], the limit is indeterminate.
Wrong. [∞] is a determinate result.
3. All limits giving indeterminate forms are undefined.
False. Many such limits exist and are finite.
4. Cancellation can be done directly.
Cancellation is valid only after proper factorisation.
5. [∞/∞] always equals 1.
Incorrect. The result depends on the highest power terms.
6. Indeterminate forms indicate mistakes.
No. They naturally arise and guide further analysis.
7. [0·∞] always equals zero.
Wrong. The product may take different values.
8. Rationalisation is used only in surds.
It is also a powerful technique for resolving limits.
9. Powers never produce indeterminate forms.
False. Forms like [0^0] and [1^∞] are common.
10. One method works for all indeterminate forms.
Different forms require different techniques.
8. Practice Questions with Step-by-Step Solutions
Question 1. Check whether the limit is an indeterminate form:
[\lim_{x \to 2} \dfrac{x^2 – 4}{x – 2}]
Step-by-Step Solution:
Substitute [x = 2] directly into the numerator:
[x^2 – 4 = 2^2 – 4 = 0]
Substitute [x = 2] into the denominator:
[x – 2 = 0]
After substitution, the expression becomes:
[\dfrac{0}{0}]
The form [0/0] does not give a definite value.
Conclusion:
The given limit is in the indeterminate form [0/0].
Question 2. Check whether the limit is an indeterminate form:
[\lim_{x \to 0} \dfrac{\sin x}{x}]
Step-by-Step Solution:
Substitute [x = 0] in the numerator:
[\sin 0 = 0]
Substitute [x = 0] in the denominator:
[x = 0]
The resulting form is:
[\dfrac{0}{0}]
Since both numerator and denominator approach zero, the result is not directly known.
Conclusion:
The limit is in the indeterminate form [0/0].
Question 3. Check whether the limit is an indeterminate form:
[\lim_{x \to 0} \dfrac{1}{x^2}]
Step-by-Step Solution:
Substitute [x = 0] into the denominator:
[x^2 = 0]
The expression becomes:
[\dfrac{1}{0}]
This result tends to infinity.
The value [∞] is a definite result, not ambiguous.
Conclusion:
The limit is NOT an indeterminate form.
Question 4. Check whether the limit is an indeterminate form:
[\lim_{x \to \infty} \dfrac{3x^2 + 1}{5x^2 – 4}]
Step-by-Step Solution:
As [x \to ∞], the numerator [3x^2 + 1] also tends to [∞].
The denominator [5x^2 – 4] also tends to [∞].
The resulting form is:
[\dfrac{∞}{∞}]
This form does not give a definite value directly.
Conclusion:
The limit is in the indeterminate form [∞/∞].
Question 5. Check whether the limit is an indeterminate form:
[\lim_{x \to 0} x \cos \dfrac{1}{x}]
Step-by-Step Solution:
Substitute [x = 0] in the first factor:
[x = 0]
As [x \to 0], the second factor [\cos (1/x)] oscillates between [-1] and [1].
So the expression behaves like:
[0 · (bounded quantity)]
This results in the form:
[0 · ∞] (ambiguous behaviour)
Conclusion:
The limit is in the indeterminate form [0·∞].
Question 6. Check whether the limit is an indeterminate form:
[\lim_{x \to \infty} (x – \sqrt{x^2 + x})]
Step-by-Step Solution:
As [x \to ∞], the term [x] tends to [∞].
The term [\sqrt{x^2 + x}] also tends to [∞].
The expression becomes:
[∞ − ∞]
This subtraction does not give a definite value directly.
Conclusion:
The limit is in the indeterminate form [∞ − ∞].
Question 7. Check whether the limit is an indeterminate form:
[\lim_{x \to 0^+} x^x]
Step-by-Step Solution:
As [x \to 0^+], the base [x] tends to [0].
The exponent [x] also tends to [0].
The resulting form is:
[0^0]
This form does not have a fixed value in limits.
Conclusion:
The limit is in the indeterminate form [0^0].
Question 8. Check whether the limit is an indeterminate form:
[\lim_{x \to \infty} \left(1 + \dfrac{1}{x}\right)^x]
Step-by-Step Solution:
As [x \to ∞], the base becomes:
[1 + \dfrac{1}{x} \to 1]
The exponent [x] tends to [∞].
The resulting form is:
[1^∞]
This form is ambiguous and needs further analysis.
Conclusion:
The limit is in the indeterminate form [1^∞].
Question 9. Check whether the limit is an indeterminate form:
[\lim_{x \to 0} (5x)]
Step-by-Step Solution:
Substitute [x = 0] directly:
[5x = 0]
The limit evaluates to a definite number.
No ambiguity is present.
Conclusion:
The limit is NOT an indeterminate form.
Question 10. Check whether the limit is an indeterminate form:
[\lim_{x \to \infty} x^{1/x}]
Step-by-Step Solution:
As [x \to ∞], the base [x] tends to [∞].
The exponent [1/x] tends to [0].
The resulting form is:
[∞^0]
This form does not give a definite value directly.
Conclusion:
The limit is in the indeterminate form [∞^0].
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