Limits & One-Sided Limits

Last lesson closed on a puzzle. The function 1/x is not defined at x = 0, yet as you creep toward zero along its curve it plainly does something — it climbs without bound.

To describe that honestly, you have to stop asking “what is the height at the point?” and start asking “where is the height headed as I approach?” Picture walking along a graph toward some x-value, watching only where you are going — the destination, not the arrival.

That “where am I headed” has a name: the limit. The gap between heading-toward and actually-arriving sounds tiny, but it is the engine behind everything ahead — derivatives, continuity, the whole Calculus block — and GATE leans on it as a quiet first step in nearly every limit and continuity question.

The same question drives real machine learning: a training loss flattening toward a floor, or a learning rate decaying toward zero, are both limits in disguise.

The intuition

Walk along the graph toward x = a and keep your eye on the height. If that height settles toward a single number L no matter how you approach, we write

lim_{x → a} f(x) = L

and read it “the limit of f as x approaches a is L.” The key word is approaches: L is decided by the neighbourhood of a — the values of f at points arbitrarily close to a on either side — and never by the single point a itself.

That last clause is the one to say out loud, because it is the confusion the whole topic rests on: the limit asks where the function is going, not where it is.

Those are separate questions with separate answers. A graph can have a hole at a and still have a perfectly good limit there; it can be defined at a to some unrelated value and the limit ignores it entirely. Whenever a limit question feels wrong, check whether you have quietly answered “what is f(a)?” instead of “where is f headed near a?”

A concrete example

Concretely: let f(x) = x + 2 everywhere except at x = 2, where somebody has declared f(2) = 7. Creep in from the left and the heights run 3.9, 3.99, 3.999; creep in from the right and they run 4.1, 4.01, 4.001.

Both sides head to 4, so lim_{x → 2} f(x) = 4 — while f(2) = 7 sits there, off the trend line, changing nothing.

To watch a limit actually happen, drag the slider that controls h in the widget below. The secant line — the straight line cutting the curve at two points — rotates as h shrinks, and as h → 0 it settles onto one fixed line.

That settling, that approach to a single position, is exactly what “the limit exists” looks like with your own eyes.

Left-hand vs right-hand limits

You can sneak up on a from two sides, and which side you choose can matter:

• Left-hand limit lim_{x → a⁻} f(x) — approach with x slightly less than a (from the left).
• Right-hand limit lim_{x → a⁺} f(x) — approach with x slightly greater than a (from the right).

The two-sided limit lim_{x → a} f(x) exists only when both one-sided limits exist and are equal. If the left says 2 and the right says 3, the curve is heading to two different places depending on direction, so there is no single destination — the limit does not exist.

Limits at infinity

Instead of approaching a finite point, we can ask where f(x) heads as x runs off to +∞ or −∞. That captures the end behaviour of the graph and pins down any horizontal asymptote.

A horizontal asymptote is a horizontal line the curve flattens toward forever without ever reaching. The classic one is

lim_{x → ∞} 1/x = 0

As x grows huge, 1/x shrinks toward 0 without ever touching it, so y = 0 is a horizontal asymptote.

The other examples are:

• Polynomials behave the opposite way, running off to ±∞.
• eˣ blows up to +∞ on the right yet decays to 0 on the left.

Each of these is just the heading-toward question pointed at the far ends of the axis.

How GATE asks this

This material rides embedded inside limit and continuity questions.

A question may:

• hand you a piecewise function — one defined by different formulas on different stretches of the x-axis — and ask whether the limit at the seam between two stretches exists;
• ask for one one-sided value; or
• ask for an at-infinity limit (typed as a NAT) and the matching asymptote (an MCQ).

You answer by checking both one-sided limits and comparing. The reflex never changes: approach from each relevant side, then compare where the sides point.

Worked example

A function is defined piecewise as f(x) = x + 1 for x < 1 and f(x) = x + 2 for x ≥ 1. Does lim_{x → 1} f(x) exist? Separately, evaluate lim_{x → ∞} 1/x.

Start with the seam at x = 1, taking each side on its own.

left  limit:  lim_{x → 1⁻} (x + 1) = 1 + 1 = 2
right limit:  lim_{x → 1⁺} (x + 2) = 1 + 2 = 3

The left-hand limit is 2 and the right-hand limit is 3. They disagree, so the curve heads to two different heights depending on the direction of approach.

Therefore the two-sided limit lim_{x → 1} f(x) does not exist — even though f(1) = 3 is perfectly well defined. Existence of the limit and the value of f(1) are simply separate questions, and the worked answer shows them parting ways.

Now the at-infinity part.

lim_{x → ∞} 1/x = 0

As x grows without bound, 1/x collapses toward 0. The value 0 is never actually attained, but that is exactly what a limit at infinity reports — where the function is headed, not where it lands.

A question to carry forward

We now know what a limit means and how to read one off a graph or sneak up on it from each side. But reading is not yet computing.

Try to evaluate lim_{x → 1} (x² − 1)/(x − 1) by the obvious move: substitute x = 1. You get 0/0, which is nonsense, not an answer. Yet the curve clearly heads somewhere definite.

Here is the thread onward: when plugging in produces a meaningless 0/0, what honest algebra recovers the value the function is really approaching?

In one breath

• A limit lim_{x → a} f(x) = L is the value f heads toward near a — decided by the neighbourhood, not by f(a).
• One-sided limits: approach from the left (a⁻) or right (a⁺). The two-sided limit exists only when both agree; disagree (2 vs 3) ⇒ does not exist.
• A limit can exist even where f(a) is undefined (a hole), and can differ from a defined f(a). lim = f(a) is the special case called continuity.
• Limits at infinity give end behaviour and horizontal asymptotes: lim_{x→∞} 1/x = 0, polynomials → ±∞, eˣ → +∞ right and 0 left.
• Reflex: approach from each relevant side and compare before declaring a limit.

Practice