L’Hôpital’s Rule & Indeterminate Forms — Study Notes

Session logistics and course updates

One-hour SI sessions available; questions about SI answered.
An SI folder exists on Brightspace for updates, including polls for session times.
Scheduling: once the other section is added, sessions’ times will be finalized.
Homework schedule:
- Homework 2 is due next week (homework indexed by week).
- Homework 1 had no assignment; due Thursday before midnight on Gradescope.
- Homework problems include ENV (engineering problems) to align MAP 2029 with engineering problems; PDF available on Brightspace → content → syllabus → download.
Quizzes: there will be a quiz every Thursday based on the homework and discussion.
Moving into a regular rhythm: weekly homework due Thursdays and quizzes on Thursdays.

We are continuing in Chapter 5, section 5.8a: L’Hôpital’s rule.
Focus: indeterminate forms zero over zero (0/0) and infinity over infinity (∞/∞).
Recap from calculus 1: certain forms like 0/0 can yield many possible limits unless you do more work; L’Hôpital’s rule helps determine the limit when applicable.
The rule is:
- If
  
  rac{f(x)}{g(x)} o ext{ indeterminate form } 0/0 ext{ or } rac{ ext{∞}}{ ext{∞}}
  
  as x o a (where a may be finite or infinite),
  ext{and } f ext{ and } g ext{ are differentiable near } a,
  
  then
  
  rac{f(x)}{g(x)} o rac{f'(x)}{g'(x)}
  as x o a, provided the limit on the right exists.
Important nuance: this is not the quotient rule; it is a separate rule that asks you to differentiate the numerator and denominator and then evaluate the limit of the new quotient.
Notation: some people write L’Hôpital’s rule as L' (or L’Ho) to emphasize differentiating; the instructor emphasizes clearly indicating when L’Hôpital has been used.
Conditions to emphasize for exams/graded work: always identify the indeterminate form before applying L’Hôpital; and distinguish between numbers, +∞, -∞, and does-not-exist when reporting limits.

Primary indeterminate forms to use L’Hôpital for (the “green” forms):
- 0/0
- ∞/∞
Cautionary forms (the “orange” forms) where L’Hôpital can still be useful but require tricks or manipulation to convert to a quotient form that is 0/0 or ∞/∞:
- ∞ · 0
- 0 · ∞
- ∞ − ∞
Exponential indeterminate forms handled with a logarithm trick:
- 0^0
- ∞^0
- 1^∞
The three “flavors” of limits discussed: numbers, +∞, -∞, and does-not-exist. These distinctions matter for full credit on exams.
A practical reminder: you can apply L’Hôpital multiple times if each subsequent application remains valid (i.e., the form stays 0/0 or ∞/∞ after each differentiation). You may need to simplify algebraically between steps.

If the limit is not initially in a 0/0 or ∞/∞ form, create one via algebraic manipulation:
- Convert products to quotients (e.g., transform ∞ · 0 to a quotient by inverting a factor).
- Use reciprocal forms to create a quotient (e.g., rewrite a product or reciprocal to form a ratio).
- Use sublimits for complex expressions (see below).
Sublimits (for complex indeterminate forms like ∞ − ∞):
- Identify a sublimit by factoring out the dominant term to isolate a part of the expression.
- Evaluate the sublimit separately (often using L’Hôpital), then infer the behavior of the original limit from the sublimit.
- Example logic: if you can write the limit as (something) / (something else) and you can show the sublimit of a portion tends to ∞, 0, or a finite value, you can deduce the overall limit after algebraic simplification.
When dealing with ∞ − ∞, a common approach is to factor out the dominant growth term and form a quotient that can be treated with L’Hôpital.
For products that resemble ∞ · 0, rewrite as a quotient (e.g., as (ln x) / (1/x) or x / (1/x)) to create a 0/0 or ∞/∞ form.
Pattern recognition for using log tricks with exponentials:
- If you have an expression of the form f(x)^{g(x)} with f(x) → 1 and g(x) → ∞, take logarithms to bring the exponent down and study the limit of the logarithm.
- Then exponentiate to recover the original limit.

Growth-rate hierarchy (useful for quick judgments about limits and time complexity):
rac{
}
- ln x grows slower than any power of x
- x grows faster than ln x:
  rac{
}{x} o 0 ext{ as } x o \n
- e^x dominates any polynomial in x: for any positive integer n,
  rac{x^n}{e^x} o 0 ext{ as } x o \n
- Therefore e^x grows much faster than any polynomial; Ln x grows slower than x, and e^x grows fastest among these basic functions.
Practical takeaway for problem solving:
- If you see a quotient with an exponential in the numerator or denominator, expect quick divergence to ∞ or 0 after a single L’Hôpital step.
- For expressions like e^x/p(x), often you’ll get ∞ after differentiation because e^x is its own derivative.
- When comparing growth rates, Ln x/x → 0, and e^x/x → ∞ as x → ∞.
Connection to broader concepts: recognizing growth rates helps anticipate when a limit will diverge or converge and informs algorithmic complexity considerations (e.g., exponential time vs polynomial time).