Cheatsheet: Limits done carefully
The limit
Section titled “The limit”lim (x->a) f(x) = the value f(x) approaches as x approaches a.
For continuous functions, just plug in: lim (x->2) x^2 = 4. The interesting cases are indeterminate forms (0/0, ∞/∞), where plug-in fails.
What “approaches” means (epsilon-delta)
Section titled “What “approaches” means (epsilon-delta)”Formal: lim (x->a) f(x) = L if for every ε > 0 there is a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.
Plain: you can force f(x) as close to L as you want by making x close enough to a. Challenge (any ε) and response (a δ that works).
Example, x^2 -> 4: |x^2 - 4| = |x-2|·|x+2| ≈ 4·|x-2|, so δ ≈ ε/4 works.
L’Hopital’s rule (for 0/0 or infinity/infinity only)
Section titled “L’Hopital’s rule (for 0/0 or infinity/infinity only)”lim (x->a) f(x)/g(x) = lim (x->a) f'(x)/g'(x)Differentiate numerator and denominator separately (not the quotient rule), then retry. Repeat if still indeterminate.
Worked examples
Section titled “Worked examples”| Limit | Form | L’Hopital | Result |
|---|---|---|---|
lim x->2 x^2 | fine | (plug in) | 4 |
lim x->0 sin(x)/x | 0/0 | cos(x)/1 | 1 (= small-angle, L4) |
lim x->0 (e^x-1)/x | 0/0 | e^x/1 | 1 (uses L7) |
lim x->0 (1-cos x)/x^2 | 0/0 | sin x/2x (still 0/0) -> cos x/2 | 1/2 (twice) |
lim x->infinity (ln x)/x | inf/inf | (1/x)/1 | 0 (x outpaces ln x) |
No limit at all: sin(1/x) as x->0 oscillates between -1 and 1 infinitely fast; epsilon-delta confirms no L works.
Why L’Hopital works
Section titled “Why L’Hopital works”Near a: f(x) ≈ f(a) + f'(a)(x-a) (first-order Taylor). When f(a)=g(a)=0, the constants vanish and f/g ≈ f'(a)(x-a)/g'(a)(x-a) = f'(a)/g'(a). The shared (x-a) cancels; the derivatives’ ratio remains. (Seed of Taylor series, L13.)
Why it matters for AI
Section titled “Why it matters for AI”Limits underwrite the field’s guarantees: convergence analysis (does gradient descent settle?), continuous-time models (the step-size -> 0 limit of an ODE solver), and universal approximation theorems (limit-based existence). Rarely computed by hand; always underneath.
Pitfalls to dodge
Section titled “Pitfalls to dodge”- L’Hopital on a non-indeterminate form. Only
0/0or∞/∞. Check first. - Using the quotient rule. L’Hopital differentiates top and bottom separately.
- Stopping at a persistent 0/0. Apply again until the form is determinate.
- Substituting too early. Simplify first, then take the limit; never set
h=0in(...)/hup front.
The one-line version
Section titled “The one-line version”A limit is what f(x) approaches (epsilon-delta: forceable arbitrarily close), and L’Hopital handles 0/0 by swapping the functions for their derivatives, keeping only the leading behavior.