MISSION · Calculus
What a Limit Is
The value a function approaches, which need not be the value it takes
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LEVEL 2 · LEARN
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The idea in one sentence
A limit says where a function is heading as the input closes in on a number — and it deliberately ignores what happens at that number.
\[\lim_{x \to a} f(x) = L\]The gap that doesn’t matter
Take \(f(x) = \frac{x^2 - 1}{x - 1}\). At \(x = 1\) it is undefined: the bottom is zero. But factor the top and cancel, and away from \(x = 1\) it is just \(x + 1\).
So the graph is the line \(y = x + 1\) with a single point punched out of it at \((1, 2)\). Approach \(x = 1\) from either side and the values head straight for 2, even though the function never takes that value.
\[\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2\]That’s the whole idea. The limit describes the journey, not the destination, and a hole in the graph is invisible to it.
From both sides
For a limit to exist, the approach from the left and the approach from the right must agree.
A step function that jumps from 2 to 5 at \(x = 3\) has a left-hand limit of 2 and a right-hand limit of 5. They disagree, so there is no limit at 3 — not an ambiguous one, no limit at all.
This is how a limit detects a break in a graph. A function is continuous at a point when three things hold: the function is defined there, the limit exists there, and the two are equal. Continuity is just “the limit is what you’d guess by pointing at the graph.”
Why bother
Because the two central ideas of calculus are both limits that can’t be evaluated directly.
A derivative is the slope between two points as the gap between them shrinks to nothing. Set the gap to zero outright and you get \(\frac{0}{0}\), which says nothing. Let it approach zero and you get a definite number.
An integral is the area under a curve as the rectangles approximating it get infinitely thin.
Both need a way to talk about “arbitrarily close to zero” without ever being zero. That’s what limits provide, and it’s why they come first in every calculus course.
Limits at infinity
You can also ask where a function heads as \(x\) grows without bound. As \(x \to \infty\), \(\frac{1}{x} \to 0\) — never arriving, but getting as close as you like. That’s a horizontal asymptote, described precisely.
Common mix-ups
- Assuming the limit equals the value. Usually it does. The interesting cases are where it doesn’t.
- Reading \(\frac{0}{0}\) as an answer. It’s a signal that more work is needed — factor, cancel, or rationalise, then take the limit.
- Ignoring one side. Both must agree for the limit to exist.
- Treating infinity as a number. It’s a direction of travel, not a destination.
Check yourself
Find \(\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\), and say whether the function is continuous at \(x = 3\).
Factoring gives \(\frac{(x-3)(x+3)}{x-3} = x + 3\) for \(x \ne 3\), so the limit is 6. The function is not continuous there — it isn’t even defined there — but the hole could be patched by declaring \(f(3) = 6\).
LEVEL 3 · REFLECT
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