What a function is heading towards, whether or not it gets there. Jumps, holes, wild oscillation, the ε–δ game, and the squeeze theorem
A limit asks where a function is heading as x closes in on a point, not what happens when it arrives. The function might be undefined there, or defined as something else entirely. The limit does not care: it only watches the approach.
Pick an example and watch the two beads close in on x = a from the left and the right. Do their heights settle on the same number?
Left limit: 2. Right limit: 2. So the limit is 2. And f(1) is undefined.
At x = 1 the formula reads 0/0, so f(1) is undefined. Everywhere else it equals x + 1, and the beads head for 2. A limit never looks at the point itself.
| distance from a | f(a − h) | f(a + h) |
|---|---|---|
| h = 10⁻¹ | 1.9 | 2.1 |
| h = 10⁻² | 1.99 | 2.01 |
| h = 10⁻³ | 1.999 | 2.001 |
| h = 10⁻⁴ | 1.9999 | 2.0001 |
| h = 10⁻⁵ | 1.99999 | 2.00001 |
| h = 10⁻⁶ | 1.999999 | 2.000001 |
An open circle marks a height the curve approaches but does not take at x = a; a filled one marks the actual value f(a). The beads close in to within 10⁻⁴ of the half-width, so for the jump and the blow-ups you can watch the two sides part company. For your own function the verdict comes from evaluating f closer and closer to a, which is evidence, not proof.
We write lim_(x → a⁻) f(x) = L for the left-hand limit, when f(x) gets as close as we like to L for all x slightly less than a, and lim_(x → a⁺) for the right-hand limit. The two-sided limit lim_(x → a) f(x) exists exactly when both one-sided limits exist and agree.
The gallery shows the standard ways this can go wrong, or right despite appearances:
The beads only provide evidence. Numerical approach can be fooled: a function could behave perfectly at every point a computer tries and still misbehave in between. Deciding a limit for certain takes algebra, inequalities, or the ε–δ definition of the next demo.
Set a tolerance ε around the claimed limit L. The demo finds how close to a you must stay, δ, to keep the graph inside the band. Shrink ε and see whether a δ keeps answering.
The largest δ is found by scanning outward from a on both sides and pinning down the first escape by bisection, so a spike narrower than the scan could slip through. The inset plots that δ against ε on logarithmic axes; red ticks along its floor mark tolerances with no δ at all. With “Zoom with ε” on, the view shrinks with ε, so a working limit looks the same at every scale while a failing one drifts out of the band.
lim_(x → a) f(x) = L means: for every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε.
Read it as a challenge and response. Someone names a tolerance ε, however small; you must answer with a distance δ that guarantees f(x) is within ε of L whenever x is within δ of a. The condition 0 < |x − a| leaves out x = a itself, which is why holes do not matter.
For f(x) = x² at a = 1 the response can be written down. If |x − 1| < δ ≤ 1 then |x + 1| < 3, so |x² − 1| = |x − 1||x + 1| < 3δ, and δ = min(1, ε/3) always works. The demo’s largest δ is bigger: exactly √(1 + ε) − 1, because the curve leaves the band first on the right, where it is steeper. To prove a limit you only need some δ, not the best one.
To show a limit fails, find a single ε for which no δ works. At the jump any ε below ½ will do. Bolzano gave a definition of continuity in this spirit in 1817 and Cauchy used ε-style arguments in his 1821 Cours d’analyse, but the fully quantified ε–δ form is due to Weierstrass, who taught it in his Berlin lectures of the 1860s.
Left: drag P to change the angle and compare the three areas. Right: zoom in on a function trapped between two others that meet at a point.
Angles are in radians: the sector’s area ½θ only holds because θ is measured as arc length on the unit circle. In degrees the limit would be π/180 instead of 1. The panels are independent; the right-hand view magnifies both axes by the same factor, so the quadratic funnels close faster than the window shrinks.
If g(x) ≤ f(x) ≤ h(x) near a, and g and h both tend to L as x → a, then f tends to L as well. Within any tolerance ε of L, a δ that works for both g and h also works for f, because f is caught between them.
For 0 < θ < π/2, the triangle OAP sits inside the sector OAP, which sits inside the triangle OAT. Their areas are ½ sin θ, ½ θ and ½ tan θ, so sin θ ≤ θ ≤ tan θ. Divide by sin θ and take reciprocals:
cos θ ≤ sin θ/θ ≤ 1.
Since cos θ → 1, the squeeze gives sin θ/θ → 1. Both sides of the inequality are even in θ, so the same holds for θ < 0. This limit is what makes the derivative of sin x equal to cos x, and it is why calculus measures angles in radians.
The same idea tames x² sin(1/x): whatever sin(1/x) does, it stays between −1 and 1, so the product stays between −x² and x², and both bounds go to 0.
Spot a mistake or have an idea?