---
title: "Define indeterminate form of a function"
subject: "Mathematics I"
chapter: 1
key: "semester_1/mathematics_i:indeterminate-form-define"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/indeterminate-form-define
site: BCT Engineering
---

# Define indeterminate form of a function

Mathematics I · Chapter 1: Derivatives and their Applications

- Asked in: 2074 Asoj
- Video: [Indeterminate Forms](https://www.youtube.com/watch?v=oEEXnyupzdo) — BriTheMathGuy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/indeterminate-form-define
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-1

## Question

Define indeterminate form of a function.

## Short answer

An indeterminate form occurs when direct substitution in a limit yields an expression like 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0^0, or ∞^0 whose value is not immediately determined. The limit may exist or not, requiring further analysis, often using L'Hospital's rule or algebraic manipulation.

## Answer

A limit lim_{x→a} f(x) is said to be in an **indeterminate form** when direct substitution of x = a gives an expression whose value cannot be decided, so the limit may be any number or may not exist, and further work is needed.

The seven standard indeterminate forms are:

```text
0/0,  ∞/∞,  0·∞,  ∞ − ∞,  1^∞,  0⁰,  ∞⁰
```
**Example:** lim_{x→0} (sin x)/x gives 0/0 on substitution but its value is 1.

The forms 0/0 and ∞/∞ are evaluated by L'Hospital's rule. The other forms are first rewritten as 0/0 or ∞/∞: 0·∞ by writing f·g = f/(1/g), ∞ − ∞ by taking a common denominator, and 1^∞, 0⁰, ∞⁰ by taking logarithms.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A limit is in indeterminate form when:
   A. direct substitution does not decide its value
   B. it is infinite
   C. it equals zero
   D. the function is undefined everywhere
   Answer: A. The value must be worked out further. (Key point: Meaning)

2. Which of these is NOT a standard indeterminate form?
   A. 1/0 positive
   B. infinity - infinity
   C. 0/0
   D. 1^infinity
   Answer: A. 1/0 is not among the seven standard forms. (Key point: Standard forms)

3. A 0 * infinity form is handled by:
   A. taking logarithms
   B. writing f g as f/(1/g)
   C. adding terms
   D. squaring
   Answer: B. This gives 0/0 or infinity/infinity. (Key point: Conversion)

4. lim sin x / x as x tends to 0 is 0/0 yet equals:
   A. infinity
   B. 0
   C. does not exist
   D. 1
   Answer: D. Shows the form alone does not fix the value. (Key point: Example)

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