Course: Calculus II
Topic: Calculus of Infinity
Subtopic: Indeterminate Forms and L'Hopital's Rule


We know that zero divided by a non-zero number is zero, and that a non-zero number divided by zero is "undefined", but what about zero divided by zero? We know that 0^1=0 and that 1^0=1, but what about 0^0? Things like 0/0 or 0^0 are called indeterminate forms because they are not necessarily determinable. We will look at them in terms of limits, such as lim(x/sin(x)) as x->0 (which appears to approach 0/0).

Early in Calculus I we studied various approaches to evaluating limits (worth reviewing now!) including those of the form 0/0. We attacked them using algebra to rewrite or simplify the function. But a nice alternative is to use L'Hopital's Rule. This rule can be used to evaluate limits of the form 0/0 or ∞/∞.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: the four "special limits" from Calculus I (shown again below), determinate form, indeterminate form, L'Hopital's Rule

Special Limits

`lim_(theta->0) sintheta/theta = 1` `lim_(theta->0) tantheta/theta = 1`
`lim_(theta->0) (1-costheta)/theta = 0` `lim_(x->0) (1+x)^(1/x) = e`

Mini-Lectures and Examples

STUDY: Calculus of Infinity - L'Hopital's Rule

Supplemental Resources (recommended)

Read and ponder this article Cantor: The Man Who Tamed Infinity

Watch this video from Numberphile Infinity is Bigger Than You Think

Watch this video from Vsauce How to Count Past Infinity

Supplemental Resources (optional)

Video: Indeterminate Forms and L'Hopital's Rule, Selwyn Hollis's Video Calculus

Lesson: L'Hopital's Rule, Dale Hoffman's Contemporary Calculus

Lesson: Types of Infinities, Paul's Online Notes

rev. 2021-02-27