Introduction to Limits: Approaching Without Arriving
f(x) = (x² - 1)/(x - 1).
What is f(1)?
Plug in: (1 - 1)/(1 - 1) = 0/0. Undefined.
But what happens as x gets close to 1?
x = 0.9: f(0.9) = (0.81 - 1)/(0.9 - 1) = -0.19/-0.1 = 1.9.
x = 0.99: f(0.99) = 1.99.
x = 0.999: f(0.999) = 1.999.
x = 1.1: f(1.1) = 2.1.
x = 1.01: f(1.01) = 2.01.
x = 1.001: f(1.001) = 2.001.
As x approaches 1, f(x) approaches 2.
That's a limit. f(1) is undefined. But the limit as x → 1 exists, and it equals 2.
The Unlock: Getting Arbitrarily Close Without Reaching
A limit describes what happens as you approach a value, not what happens at the value.
lim (x → a) f(x) = L means:
As x gets arbitrarily close to a, f(x) gets arbitrarily close to L.
You don't need f(a) to be defined. You don't need to reach a. You just need to get close.
Limits are about behavior near a point, not at the point.
Notation: lim (x → a) f(x) = L
Read: "The limit of f(x) as x approaches a is L."
x → a: x approaches a.
f(x) → L: f(x) approaches L.
The limit L is the value f(x) gets close to, not necessarily the value f(x) equals.
Intuitive Definition
lim (x → a) f(x) = L if:
You can make f(x) as close to L as you want by making x sufficiently close to a (but not equal to a).
This is informal, but it captures the idea.
Example: A Removable Discontinuity
f(x) = (x² - 1)/(x - 1).
Factor the numerator: (x - 1)(x + 1)/(x - 1).
Cancel: f(x) = x + 1 (for x ≠ 1).
For all x except 1, f(x) = x + 1.
As x → 1, f(x) → 2.
lim (x → 1) f(x) = 2.
Even though f(1) is undefined, the limit exists.
Example: A Simple Polynomial
f(x) = x² + 3.
lim (x → 2) f(x) = ?
As x → 2, f(x) → 2² + 3 = 7.
lim (x → 2) f(x) = 7.
For continuous functions, the limit is just the value at the point: lim (x → a) f(x) = f(a).
One-Sided Limits
Sometimes the function approaches different values from the left and the right.
Left-hand limit: lim (x → a⁻) f(x) = L.
Read: "The limit as x approaches a from the left."
Right-hand limit: lim (x → a⁺) f(x) = L.
Read: "The limit as x approaches a from the right."
If the left-hand and right-hand limits are equal, the limit exists:
lim (x → a) f(x) = L if and only if lim (x → a⁻) f(x) = lim (x → a⁺) f(x) = L.
Example: One-Sided Limits
f(x) = |x|/x.
For x > 0: f(x) = x/x = 1.
For x < 0: f(x) = -x/x = -1.
At x = 0, f is undefined.
lim (x → 0⁺) f(x) = 1. (Approaching from the right.)
lim (x → 0⁻) f(x) = -1. (Approaching from the left.)
The one-sided limits are different, so lim (x → 0) f(x) does not exist.
Infinite Limits
Sometimes a function grows without bound as x approaches a.
lim (x → a) f(x) = ∞ means:
As x → a, f(x) grows larger and larger without bound.
Example: f(x) = 1/x².
As x → 0, f(x) → ∞.
lim (x → 0) 1/x² = ∞.
Note: ∞ is not a number. Saying the limit is ∞ means the limit does not exist in the usual sense, but f(x) increases without bound.
Limits at Infinity
You can also take limits as x → ∞ or x → -∞.
lim (x → ∞) f(x) = L means:
As x gets arbitrarily large, f(x) approaches L.
Example: f(x) = 1/x.
As x → ∞, 1/x → 0.
lim (x → ∞) 1/x = 0.
Example: f(x) = x².
As x → ∞, x² → ∞.
lim (x → ∞) x² = ∞.
Properties of Limits
If lim (x → a) f(x) = L and lim (x → a) g(x) = M, then:
Sum: lim (x → a) [f(x) + g(x)] = L + M.
Difference: lim (x → a) [f(x) - g(x)] = L - M.
Product: lim (x → a) [f(x) · g(x)] = L · M.
Quotient: lim (x → a) [f(x) / g(x)] = L / M (if M ≠ 0).
Constant multiple: lim (x → a) [c · f(x)] = c · L.
These rules let you compute limits of combinations.
Example: Using Limit Properties
lim (x → 3) [(x² + 2x) / (x - 1)].
Numerator limit: lim (x → 3) (x² + 2x) = 9 + 6 = 15.
Denominator limit: lim (x → 3) (x - 1) = 2.
Quotient: 15 / 2.
lim (x → 3) [(x² + 2x) / (x - 1)] = 15/2.
Indeterminate Forms
Sometimes direct substitution gives an undefined form.
0/0: Could be anything. Need to simplify or use L'Hôpital's rule.
∞/∞: Could be anything. Need techniques to resolve.
0 · ∞, ∞ - ∞, 0⁰, 1^∞, ∞⁰: All indeterminate. Need further analysis.
Example: lim (x → 0) (sin(x)/x).
Direct substitution: 0/0. Indeterminate.
But using calculus or a known result: lim (x → 0) (sin(x)/x) = 1.
Squeeze Theorem
If f(x) ≤ g(x) ≤ h(x) for all x near a, and:
lim (x → a) f(x) = lim (x → a) h(x) = L,
then lim (x → a) g(x) = L.
g(x) is "squeezed" between f(x) and h(x), so it must approach the same limit.
Example: -1 ≤ sin(x) ≤ 1 for all x.
Multiply by x²: -x² ≤ x² sin(x) ≤ x².
As x → 0, -x² → 0 and x² → 0.
By the squeeze theorem: lim (x → 0) x² sin(x) = 0.
Limits and Continuity
A function f is continuous at x = a if:
- f(a) is defined.
- lim (x → a) f(x) exists.
- lim (x → a) f(x) = f(a).
If all three conditions hold, the function has no jump, gap, or hole at a.
Polynomials, exponentials, sine, cosine—these are continuous everywhere they're defined.
Rational functions are continuous except where the denominator is zero.
Why Limits Matter: The Foundation of Calculus
Calculus is built on limits.
Derivative: The derivative is the limit of the difference quotient:
f'(a) = lim (h → 0) [f(a + h) - f(a)] / h.
This measures instantaneous rate of change.
Integral: The integral is the limit of Riemann sums:
∫[a to b] f(x) dx = lim (n → ∞) Σ f(xᵢ) Δx.
This measures total accumulation.
Without limits, there's no calculus.
Example: Derivative as a Limit
f(x) = x². Find f'(2).
f'(2) = lim (h → 0) [f(2 + h) - f(2)] / h.
f(2 + h) = (2 + h)² = 4 + 4h + h².
f(2) = 4.
f'(2) = lim (h → 0) [(4 + 4h + h²) - 4] / h = lim (h → 0) (4h + h²) / h = lim (h → 0) (4 + h).
As h → 0, 4 + h → 4.
f'(2) = 4.
The derivative at x = 2 is 4.
Limits and Asymptotes
Limits describe asymptotic behavior.
Vertical asymptote at x = a: lim (x → a⁺) f(x) = ±∞ or lim (x → a⁻) f(x) = ±∞.
Horizontal asymptote y = L: lim (x → ∞) f(x) = L or lim (x → -∞) f(x) = L.
Example: f(x) = 1/(x - 2).
Vertical asymptote at x = 2: lim (x → 2⁺) f(x) = ∞, lim (x → 2⁻) f(x) = -∞.
Horizontal asymptote y = 0: lim (x → ±∞) f(x) = 0.
Limits formalize asymptotic behavior.
Evaluating Limits: Techniques
1. Direct substitution: If f is continuous at a, then lim (x → a) f(x) = f(a).
2. Factoring and canceling: Simplify the expression to remove indeterminate forms.
Example: lim (x → 2) (x² - 4)/(x - 2) = lim (x → 2) (x + 2) = 4.
3. Rationalizing: Multiply by a conjugate to eliminate square roots.
Example: lim (x → 0) (√(x + 1) - 1)/x.
Multiply by (√(x + 1) + 1)/(√(x + 1) + 1):
lim (x → 0) [(x + 1) - 1] / [x(√(x + 1) + 1)] = lim (x → 0) x / [x(√(x + 1) + 1)] = lim (x → 0) 1 / (√(x + 1) + 1) = 1/2.
4. L'Hôpital's rule (calculus): If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x).
5. Squeeze theorem: Bound the function between two simpler functions.
Limits and Graphs
Graphically, lim (x → a) f(x) = L means:
As you trace the graph from left and right toward x = a, the y-values approach L.
There might be a hole at x = a. Or the function might be defined there but equal something else. The limit is about the approach, not the arrival.
Precise Definition: Epsilon-Delta
The formal definition of a limit (from real analysis):
lim (x → a) f(x) = L means:
For every ε > 0, there exists δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
Translation: You can make f(x) arbitrarily close to L (within ε) by making x sufficiently close to a (within δ).
This is the rigorous foundation, but the intuitive idea is usually sufficient.
Why Limits Are Hard
Limits feel abstract because you never reach the value you're approaching.
You're reasoning about what would happen if you got arbitrarily close, without actually getting there.
It's a mental leap: from computation (plug in values) to reasoning about behavior (approach values).
But that leap is essential. It's the shift from arithmetic to analysis.
Limits and Infinity
Infinity is not a number. It's a concept.
lim (x → ∞) f(x) = L doesn't mean "f(∞) = L." It means "f(x) gets closer to L as x gets larger and larger."
Similarly, lim (x → a) f(x) = ∞ doesn't mean "f(a) = ∞." It means "f(x) grows without bound as x approaches a."
Infinity is a direction, not a destination.
Limits in Science and Engineering
Limits model processes that approach equilibrium or boundary conditions.
Physics: The speed of light is a limit. As objects accelerate, their speed approaches c but never reaches it.
Chemistry: Equilibrium is the limit of a reaction as time goes to infinity.
Economics: Marginal cost is the limit of the difference in cost as production increases by an infinitesimal amount.
Limits formalize the idea of "getting arbitrarily close."
Common Mistakes
Mistake 1: Confusing lim (x → a) f(x) with f(a).
The limit is the value f(x) approaches, not the value it equals at a.
Mistake 2: Assuming limits always exist.
If left-hand and right-hand limits differ, the limit doesn't exist.
Mistake 3: Treating ∞ as a number.
∞ is not a number. It's shorthand for "grows without bound."
Mistake 4: Direct substitution for indeterminate forms.
0/0, ∞/∞, etc. require further analysis. Don't just plug in.
Mistake 5: Forgetting to check one-sided limits.
For piecewise functions or functions with jumps, check left and right separately.
Why Limits Are the Gateway to Calculus
Precalculus gives you functions. Calculus asks: How do they change?
But "change" is tricky. The instantaneous rate of change is a limit. The area under a curve is a limit.
Everything in calculus is built on limits.
If you understand limits—really understand them—calculus becomes clear. If you don't, calculus is a blur of formulas.
Limits are the conceptual foundation. Master them, and everything else follows.
The Payoff: Reasoning About the Unreachable
When you understand limits, you can reason about things you can't directly compute.
You can talk about the behavior of a function at a point where it's undefined.
You can talk about what happens as time goes to infinity, or as a quantity shrinks to zero.
Limits let you reason about the edge cases—the boundaries, the extremes, the infinitesimal.
That's the power: you're no longer confined to what you can plug in and calculate. You can reason about what happens in the limit.
And that's the shift from algebra to analysis. From computing to understanding.
Part 10 of the Precalculus series.
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