Limits and continuity

The formula

lim (x → a) f(x) = L

Reading: as x gets arbitrarily close to a (from either side, but not necessarily equal to a), f(x) gets arbitrarily close to L.

For every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ  ⟹  |f(x) − L| < ε

Reading: name any error band ε you want around L; there is always a window δ around a tight enough that every x inside it lands inside your band. This is the definition that makes "close" precise.

f is continuous at a   ⟺   lim (x → a) f(x) = f(a)

Reading: three things must all be true and agree — f(a) exists, the limit exists, and they are equal. No hole, no jump, no spike.

L'Hôpital's rule:  if f/g → 0/0 or ∞/∞,  then  lim f(x)/g(x) = lim f'(x)/g'(x)

Reading: when a quotient collapses into a meaningless 0/0 or ∞/∞ at the point, replace top and bottom with their derivatives and take the limit again. Repeat if it stays indeterminate.

Where you meet it

Op-amp gain on the bench. The non-inverting closed-loop gain is G = A / (1 + A·β), where A is the open-loop gain and β = R1/(R1+R2) is the feedback fraction. A real op-amp has A around 10⁵ to 10⁶, not infinity — but the design formula G = 1 + R2/R1 is the limit as A → ∞. With R1 = 1 kΩ, R2 = 9 kΩ, β = 0.1: A = 10⁵ gives 9.9990, A = 10⁷ gives 9.99999, and the limit is exactly 10. That's why the datasheet gain-bandwidth number matters: you're always operating a finite distance from the idealized limit.
RC filter at the corner. The first-order lowpass magnitude is |H| = 1 / √(1 + (ω/ω_c)²). Walk ω up to the cutoff ω_c and the expression stays perfectly continuous — it passes through 1/√2 = 0.7071, which is −3.01 dB, with no discontinuity. Continuity is why you can sweep a signal generator across the corner and watch the response slide smoothly instead of stepping.
Sensitivity and gain formulas that go 0/0. Transfer functions and sensitivity ratios routinely produce 0/0 at a specific frequency or operating point — a zero cancels a pole, a numerator and denominator both vanish. On a design review, someone plugs in the singular value, the calculator throws NaN, and the question becomes: is that a real singularity, or a removable hole where the true value is finite?
Small-angle and small-signal work. Every sin(x) ≈ x linearization on a test stand is the limit lim (x→0) sin(x)/x = 1 doing quiet work. At x = 0.01 rad the ratio is 0.99998 — the linearization error is the distance from the limit, and it's how you justify treating a gimbal or an accelerometer response as linear near null.

How it works

A limit describes approach, not arrival. lim (x→a) f(x) deliberately says nothing about f(a) itself — the 0 < |x − a| in the definition excludes the point. That's the whole trick behind resolving 0/0: sin(x)/x is genuinely undefined at x = 0 (you'd be dividing zero by zero), but the limit is a clean 1. The function has a hole you can fill. That's a removable discontinuity, and it's the good case.

The bad cases are jumps and blow-ups. A jump discontinuity is a relay contact snapping, a comparator flipping — the left-hand limit and right-hand limit both exist but disagree, so no single limit exists. An infinite discontinuity is a pole: 1/x at x = 0 has no finite limit from either side. Continuity fails, and you cannot linearize, differentiate, or trust an interpolation across that point. Knowing which kind of break you have tells you whether the operating point is merely awkward (removable) or physically off-limits (pole).

L'Hôpital's rule is the workhorse for the removable case, and it's the one people misuse. Three failure modes worth burning in:

It only applies to 0/0 or ∞/∞. Apply it to a quotient that is not indeterminate — say the top goes to 3 and the bottom to 2 — and you'll get a wrong answer with total confidence. Check the form first, every time.
The derivative limit has to exist. If f'/g' oscillates or diverges, the rule tells you nothing — the original limit may still exist and be finite. A non-result from L'Hôpital is not a proof the limit fails.
It can loop. Some quotients cycle back to the same indeterminate form no matter how many times you differentiate (ratios of exponentials and radicals do this). Then you switch tools — factor, rationalize, or expand in a series instead.

Worked case, verified numerically: lim (x→0) (1 − cos x)/x². Both parts go to 0, so it's 0/0. Differentiate top and bottom: sin(x)/(2x), still 0/0. Again: cos(x)/2 → 1/2. Check it directly — at x = 0.001 the raw ratio is 0.49999996. The limit is exactly 1/2, and the numerics confirm it.

The engineer's mental model: a limit answers "where is this heading," continuity answers "does it actually get there cleanly," and L'Hôpital answers "the formula broke at the point — what's the real value hiding under the 0/0." Real hardware never sits at the singular point; it sits a measurable distance away, and the limit is the number the design converges to as that distance shrinks.

History

The rigorous machinery came in backwards — the rule got famous a century before anyone could state a limit precisely. In the 1690s the new differential calculus was spreading out of the Leibniz camp, and a French marquis, Guillaume de l'Hôpital (1661–1704), wanted to master it. He struck a paid arrangement with Johann Bernoulli: in a proposal dated March 17, 1694, l'Hôpital offered Bernoulli a retainer of 300 pounds a year in exchange for lessons in the new calculus and for Bernoulli's discoveries, which l'Hôpital could use as he pleased ^[1]^[2]. Bernoulli worked out the rule for the 0/0 form and sent it to l'Hôpital in a letter dated July 22, 1694 ^[2]^[3].

In 1696 l'Hôpital published Analyse des infiniment petits pour l'intelligence des lignes courbes — the first textbook ever written on differential calculus — and the rule appeared in it, which is why it carries his name to this day ^[1]^[2]. After l'Hôpital died in 1704, Bernoulli publicly claimed the result, and most of the book, as his own. The dispute stood on his word for two centuries until a manuscript of his 1691–92 lectures surfaced at Basel in 1921, matching l'Hôpital's text closely enough to back Bernoulli's account ^[1]^[3].

The definition underneath the rule took much longer. Bernard Bolzano gave an ε-δ style definition of continuity in 1817, but his work went largely unread in his lifetime ^[4]^[5]. Augustin-Louis Cauchy built limits into the foundations of analysis in his 1821 Cours d'analyse, defining continuity as small changes in x producing small changes in y, though he never wrote the formal ε-δ statement ^[4]^[6]. It fell to Karl Weierstrass (1815–1897), the man usually called the father of modern analysis, to put the ε-δ definition into the crisp form still taught today ^[4]^[6]. So the tool engineers reach for came first, and the rigor that justifies it arrived a hundred and twenty years later.

Related tools

/tools/opamp-gain
/tools/inamp-gain
/tools/rc-filter
/tools/rc-charge-time
/tools/series-rlc-impedance

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →