Limits of real functions
The limit asks what value a function approaches near a point, regardless of what happens exactly at that point. From that distinction come the derivative, the integral and steady-state behavior.
01What happens near the point, not at the point
The limit answers a very precise question: what value does f(x) approach as x approaches a number a, without ever reaching it. What happens exactly at x = a plays no part; the function may not even be defined there.
f(x) = (x² − 1)/(x − 1) does not exist at x = 1: it gives 0/0. But for every x ≠ 1 it can be simplified, because x² − 1 = (x − 1)(x + 1), and what remains is f(x) = x + 1.
So, as x approaches 1, the function approaches 2. This is written limx→1 f(x) = 2, and it does not matter that f(1) does not exist.
That distinction, “near” versus “at”, is what makes it possible to talk about instantaneous velocity, about the slope of a curve and about steady state. All of calculus rests on it.
02The formal definition
“Approaching” is not a mathematical phrase. The rigorous version, due to Cauchy and Weierstrass, turns the approach into a challenge between two tolerances:
One player proposes a tolerance ε —“I want the output within ±0.01”— and the other has to answer with a δ —“then keep the input within ±0.003”—. If there is an answer for every demand, the limit is L. It is the same logic as an engineering specification: how much the input can vary for the output to stay within tolerance.
The 0 < |x − a| on the left is the detail that excludes the point a itself. That is why the limit can exist even if the function is not defined there, or if it takes some other value.
Choose a function and shrink ε. If for every ε you can find a δ that works, the limit exists.
03One-sided limits
You can reach a point from the left or from the right, and the result can differ:
The typical case is the step: a signal that goes from 0 to 5 V has a limit of 0 from the left and 5 from the right, so at the instant of the edge the limit does not exist. It is not a mathematical pathology: it is exactly what you see on the oscilloscope.
04How they are computed in practice
If the limits of f and g exist and are finite, the limit of the sum is the sum, that of the product is the product and that of the quotient is the quotient as long as the denominator does not tend to zero. With that, most limits are solved by substitution: if the function is continuous at a, the limit is f(a).
The work comes with the indeterminate forms, which are the combinations where algebra alone is not enough:
| Indeterminate form | How to get out of it | Example |
|---|---|---|
| 0/0 | Factor and simplify, or rationalize | \( \dfrac{x^2 - 4}{x - 2} \to x + 2 \to 4 \) |
| ∞/∞ | Divide numerator and denominator by the highest power | \( \dfrac{3x^2 + x}{5x^2 - 1} \to \dfrac{3}{5} \) |
| ∞ − ∞ | Factor out a common term or multiply by the conjugate | \( \sqrt{x^{2} + x} - x \to \dfrac{1}{2} \) |
| 0 · ∞ | Rewrite as a quotient | \( x \ln x \) as \( x \to 0^+ \), tends to 0 |
| 1∞ | Reduce to the notable limit for the number e | (1 + 1/x)x → e |
And two notable limits worth memorizing, because they appear all the time:
05Infinite limits and asymptotes
| Situation | Meaning | On the graph |
|---|---|---|
| limx→a f(x) = ±∞ | Grows without bound as x approaches a | Vertical asymptote at x = a |
| limx→±∞ f(x) = L | Settles at L in the long run | Horizontal asymptote y = L |
| limx→∞ [f(x) − (mx + h)] = 0 | Hugs a slanted line | Oblique asymptote |
For a rational function, behavior at infinity is decided by comparing degrees: if the numerator’s is smaller, the horizontal asymptote is y = 0; if they are equal, it is the quotient of the leading coefficients; if the numerator exceeds the denominator by one degree, there is an oblique asymptote.
This is exactly what is done when drawing a Bode plot: the transfer function is studied for ω → 0 and for ω → ∞, and the asymptotes give the lines of −20 dB per decade.
06Limits you can measure with the oscilloscope
- Steady state. A charging capacitor follows v(t) = V(1 − e−t/RC). The limit as t → ∞ is V: the final voltage. In practice it is accepted that it has “arrived” after five time constants, when less than 1% remains.
- Gain of a feedback amplifier. With open-loop gain A and feedback β, the gain is A/(1 + Aβ). The limit as A → ∞ is 1/β: that is why the ideal op amp is analyzed as if its gain were infinite, and why the final gain is set by the resistors.
- Asymptotic behavior of a filter. For ω → 0 a low-pass filter lets everything through and for ω → ∞ it attenuates; the asymptotes of that limit are the lines of the Bode plot.
- Edges. At an ideal edge the left-hand and right-hand limits differ: the limit does not exist. No real circuit switches in zero time, and that is precisely the difference between the model and the oscilloscope.
07In the lab
For f(x) = (√(x + 4) − 2)/x, build a table with x = ±0.1; ±0.01; ±0.001 and conjecture the limit at 0. Then compute it by rationalizing and compare. (It gives 1/4.)
Charge a capacitor with a step and use the oscilloscope to measure the time to reach 63%, 95% and 99% of the final value. Check that they correspond to 1, 3 and 5 time constants, and discuss why the final value is a limit and not a value that is ever reached.
Measure the response of an RC low-pass filter at several frequencies, plot it on a logarithmic scale and draw the two asymptotes. Check that they cross at the cutoff frequency. You can practice this in the instrument simulator.
08Common mistakes
- Substituting and stopping at 0/0 as if it were a result: it is an indeterminate form, and it has to be worked out.
- Confusing f(a) with the limit: they can differ, and the limit can exist without f(a) existing.
- Forgetting the one-sided limits in piecewise functions or functions with an absolute value.
- Writing ∞ as if it were a number and “simplifying” ∞/∞.
- Using sin x ≈ x with x in degrees. The notable limit holds in radians.
- Believing an asymptote cannot be crossed: a horizontal one can be crossed many times before it settles.
09Self-assessment
Compute limx→3 (x² − 9)/(x − 3).
Factoring: (x − 3)(x + 3)/(x − 3) = x + 3 → 6.
limx→∞ (2x³ − x)/(5x³ + 4x²).
Dividing everything by x³: 2/5.
Does the limit at 0 of f(x) = |x|/x exist?
No: from the right it gives 1 and from the left −1. The one-sided limits exist but do not coincide.
limx→0 sin(5x)/x.
Rearrange it into the notable limit: 5·sin(5x)/(5x) → 5.
What asymptotes does f(x) = (2x + 1)/(x − 3) have?
Vertical at x = 3 and horizontal at y = 2, the quotient of the leading coefficients.
A capacitor charges to 12 V with τ = 2 ms. What is the limit of v(t) and what is its value at 10 ms?
The limit is 12 V. At 10 ms, which is 5τ, v = 12(1 − e−5) ≈ 11.92 V: 99.3%.
10Further reading
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 2 develops the limit with tables, graphs and the ε–δ definition, in that order.
- Hebe Rabuffetti. Introducción al análisis matemático (Cálculo 1) (in Spanish). El Ateneo. Treats one-sided limits and indeterminate forms with the notation used in Argentine universities.
- Michael Spivak. Calculus. 3rd ed. (Spanish edition, “Calculus”), Reverté, 2012. For anyone who wants complete rigor: the definition of the limit proved in full detail.