Limits: How to Measure Something Infinitely Small
A limit asks: what value is this approaching?
Not what value it reaches. What value it gets arbitrarily close to. You never arrive, but you can get as close as you want.
Here's the unlock: limits let you answer questions that direct calculation can't. What's 0/0? Undefined. But what's (x²-1)/(x-1) as x approaches 1? That has an answer — the limit is 2. The limit sees where the function is headed, even when direct substitution fails.
Limits are calculus's way of extracting meaningful answers from expressions that look impossible.
The Intuition
Imagine walking toward a wall. You cover half the remaining distance with each step.
You never reach the wall. But you approach it. Given any tiny gap, you can get closer than that gap.
The limit is the wall — what you're approaching, not what you reach.
In notation: lim[n→∞] (distance to wall) = 0
Formal Definition
lim[x→a] f(x) = L means:
For any tiny tolerance ε > 0, there exists some δ > 0 such that whenever 0 < |x - a| < δ, we have |f(x) - L| < ε.
Translation: No matter how close to L you demand the output be (ε), I can find inputs close enough to a (within δ) that guarantee it.
The function doesn't need to equal L at a. It needs to approach L arbitrarily closely as x approaches a.
Example: Why 0/0 Can Have a Limit
Consider f(x) = (x² - 1)/(x - 1)
At x = 1: f(1) = 0/0. Undefined.
But factor the numerator: (x² - 1)/(x - 1) = (x+1)(x-1)/(x-1) = x + 1 (when x ≠ 1)
As x approaches 1, x + 1 approaches 2.
So: lim[x→1] (x² - 1)/(x - 1) = 2
The function has a "hole" at x = 1, but the limit exists. The limit sees past the hole to where the function is heading.
One-Sided Limits
Sometimes behavior differs from left and right.
Left-hand limit: lim[x→a⁻] f(x) — approaching a from below
Right-hand limit: lim[x→a⁺] f(x) — approaching a from above
The two-sided limit exists only if both one-sided limits exist and are equal.
Example: f(x) = |x|/x
- From the right (x > 0): |x|/x = 1
- From the left (x < 0): |x|/x = -1
- lim[x→0] |x|/x does not exist (sides disagree)
Limits at Infinity
What happens as x gets arbitrarily large?
lim[x→∞] 1/x = 0
As x grows, 1/x shrinks toward zero. It never reaches zero, but it approaches it.
lim[x→∞] (3x² + 2x)/(x² + 1) = 3
Divide numerator and denominator by x². As x→∞, the lower-order terms vanish, leaving 3/1 = 3.
Infinite Limits
What if the function grows without bound?
lim[x→0⁺] 1/x = +∞
As x approaches 0 from the right, 1/x grows larger and larger. It doesn't approach any finite value — it "approaches infinity."
Technically, the limit doesn't exist (infinity isn't a number). But we write = ∞ to describe the unbounded behavior.
Limit Laws
Limits obey arithmetic rules:
Sum: lim(f + g) = lim f + lim g
Product: lim(f · g) = lim f · lim g
Quotient: lim(f/g) = lim f / lim g (if lim g ≠ 0)
Power: lim(f^n) = (lim f)^n
Constant: lim(c · f) = c · lim f
These let you break complex limits into simpler pieces.
Indeterminate Forms
Some expressions look undefined but have limits:
0/0: Might be any value. Factor, simplify, or use L'Hôpital's rule.
∞/∞: Compare growth rates. Could be 0, finite, or ∞.
0 · ∞: Rewrite as 0/(1/∞) = 0/0 or ∞/(1/0) = ∞/∞.
∞ - ∞: Could be anything. Combine fractions or factor.
1^∞, 0⁰, ∞⁰: Require careful analysis (often use logarithms).
"Indeterminate" means you can't tell from the form alone — more work is needed.
Continuity
A function is continuous at a point if:
- f(a) exists
- lim[x→a] f(x) exists
- lim[x→a] f(x) = f(a)
Translation: the function's value equals its limit. No holes, no jumps.
Continuous functions let you substitute directly: lim[x→a] f(x) = f(a).
Polynomials, trig functions, exponentials — all continuous on their domains.
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L.
The function is "squeezed" between two functions that both approach L.
Classic example: lim[x→0] x·sin(1/x) = 0
Since -|x| ≤ x·sin(1/x) ≤ |x|, and both bounds approach 0, so does the middle.
The Fundamental Limit
lim[x→0] sin(x)/x = 1 (x in radians)
This limit underlies all of calculus with trigonometry. It's why d/dx(sin x) = cos x.
You can't substitute (0/0), but geometrically, sin(x) and x become nearly equal for small x.
Why Limits Matter for Calculus
The derivative is a limit: f'(x) = lim[h→0] (f(x+h) - f(x))/h
The integral is a limit: ∫f(x)dx = lim[n→∞] Σf(xᵢ)Δx
Limits are the foundation. They let calculus work with infinitely small quantities without contradictions.
The Core Insight
Limits extract answers from the impossible.
You can't divide by zero, but you can see what a ratio approaches as the denominator shrinks. You can't evaluate at a discontinuity, but you can see where the function is heading.
The limit is the value you're approaching, not the value you reach. This distinction — between arriving and approaching — is what makes calculus possible.
When direct calculation fails, limits succeed by asking a different question: not "what is the value?" but "what value are we approaching?"
Part 2 of the Calculus Derivatives series.
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