1. What Indeterminate Forms Mean
An indeterminate form appears when a limit expression looks like it has a clear value, but actually does not. The expression hides important behaviour that cannot be resolved by direct substitution.
In simple words: the form looks settled, but the actual limit depends on how the function approaches that form.
2. Why Direct Substitution Fails
When we plug in a value while evaluating a limit, sometimes we get forms that do not reveal the actual behaviour of the function. Different functions can produce the same indeterminate form but have different limits.
3. Common Indeterminate Forms
These forms are considered indeterminate because they do not give any information about the true limit.
3.1. 1. 0/0 Form
\lim_{x \to a} \frac{0}{0}
Most common form. Needs algebraic simplification.
3.2. 2. ∞/∞ Form
\lim_{x \to \infty} \frac{\infty}{\infty}
Occurs when both numerator and denominator grow without bound.
3.3. 3. 0 · ∞ Form
0 \cdot \infty
A small number multiplied by a very large one — behaviour is uncertain.
3.4. 4. ∞ − ∞ Form
\infty - \infty
Two large values may cancel or not, depending on how fast they grow.
3.5. 5. 0<sup>0</sup> Form
0^0
Appears in exponential limits; cannot assume value 1.
3.6. 6. ∞<sup>0</sup> Form
\infty^0
Base grows unbounded while power shrinks to 0.
3.7. 7. 1<sup>∞</sup> Form
1^\infty
A very subtle form used in growth and decay problems.
4. Examples Showing Indeterminate Behaviour
These examples show how the same form can lead to different limits.
4.1. Example 1 — 0/0 Form
\lim_{x \to 2} \frac{x^2 - 4}{x - 2}
Direct substitution gives 0/0. But simplifying:
\frac{(x-2)(x+2)}{x-2} = x + 2
\lim_{x \to 2} (x + 2) = 4
4.2. Example 2 — ∞/∞ Form
\lim_{x \to \infty} \frac{3x^2 - 1}{6x^2 + 4}
Leading terms dominate → limit is 1/2.
4.3. Example 3 — ∞ − ∞ Form
\lim_{x \to \infty} (\sqrt{x^2 + 1} - x)
Even though both terms grow to ∞, the difference approaches 0 after rationalizing.
5. Why These Forms Are Called 'Indeterminate'
The same form can lead to different limits depending on the function. For example:
5.1. 0/0 Examples
- \( \lim_{x \to 0} \frac{x}{x} = 1 \)
- \( \lim_{x \to 0} \frac{x^2}{x} = 0 \)
- \( \lim_{x \to 0} \frac{|x|}{x} \) does not exist
6. How Indeterminate Forms Are Usually Resolved
To evaluate limits that produce indeterminate forms, common techniques include:
6.1. Algebraic Methods
- Factoring
- Rationalizing
- Simplifying expressions
6.2. Using Special Identities
Manipulating expressions to reveal hidden behaviour.
6.3. L'Hospital's Rule
Used mainly for 0/0 and ∞/∞ forms by differentiating numerator and denominator.
7. Why Understanding Indeterminate Forms Is Important
They highlight situations where limit behaviour is subtle and cannot be guessed. Many calculus techniques are built around resolving these forms correctly.