Limits & One-Sided Limits

Picture yourself walking along a graph toward some x-value. You’re not asking “what’s the height at that point” — you’re asking “where am I headed?” That’s a limit. The destination, not the arrival.

The distinction sounds tiny, but it’s the engine behind everything that comes next — derivatives, continuity, the whole Calculus block. GATE leans on it as a quiet first step in almost every limit and continuity question. The same “where is this heading?” question drives real ML too: a training loss curve flattening toward a floor, or a learning rate decaying toward zero, are both limits in disguise.

The intuition

Imagine walking along the graph toward x = a and watching the height. If the height settles toward a single number L no matter how you approach, we write

lim_{x → a} f(x) = L

and say “the limit of f as x approaches a is L”. Crucially, L is about the neighbourhood of a, not the single point a. The function might have a hole there, or be defined to some unrelated value — the limit ignores that and reports where the curve is pointing.

To watch a limit happen, drag the slider that controls h in the widget below. The secant line (the straight line cutting through two points on the curve) rotates as h shrinks, and in the limit h → 0 it settles onto a single tangent. That settling is what “the limit exists” looks like geometrically.

Left-hand vs right-hand limits

You can sneak up on a from two sides:

• 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 function is heading to two different places depending on direction, so there is no single limit — it 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 −∞. This captures the end behavior of the graph and pins down any horizontal asymptote. The classic is

lim_{x → ∞} 1/x = 0

— as x grows huge, 1/x shrinks toward (but never reaches) 0, so y = 0 is a horizontal asymptote. Polynomials, by contrast, run off to ±∞; eˣ blows up to +∞ on the right but heads to 0 on the left.

How GATE asks this

This material is embedded in limit and continuity questions. A question may hand you a piecewise function and ask whether the limit at the seam exists (you check both one-sided limits), or ask for a one-sided limit value, or for an at-infinity limit (asked as a NAT — type the number) and an asymptote (asked as an MCQ). The reflex is always: approach from each relevant side and compare.

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.

Check the two sides at x = 1.

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 function is heading 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 separate questions.)

The at-infinity limit.

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 — the value the function approaches.

Quick check