Limit of a Function

A limit asks where a function is headed as you push x arbitrarily close to some value. Remarkably, x never has to actually arrive there, and the function need not even be defined at that spot. Even at a punched-out hole, if both sides approach the same value, the limit still exists for certain. Drag the closeness slider and watch the blue point on the left and the orange point on the right both close in on 2, right where the hole sits.

What is a Limit?

🎯 Core Idea of a Limit

①"As x gets arbitrarily close to a, what does f(x) approach?"

②No need to actually reach x = a — the trend matters

③Left and right limits must agree for the limit to exist

Closeness t20 %

💡 What to Observe

①Blue dot (left limit) and orange dot (right limit) both approach y = 2

②f(1) is undefined (open circle) but the limit is 2

③Factoring (x²−1)/(x−1) = x+1 — close to x=1 means f(x) is close to 2

Epsilon–Delta Definition

ε (tolerance)1.5

Definition (ε-δ)

lim_x→a f(x) = L

For every ε > 0 there exists δ > 0 with 0 < |x−a| < δ ⇒ |f(x) − L| < ε

🔑 How to Read ε-δ

①ε is the allowed error along the y-axis (orange band)

②δ is the constraint along the x-axis (blue band)

③If a δ exists for every ε, however small, the limit exists

Basic Limit Properties

Constant k2

Arithmetic of Limits

lim [f(x) ± g(x)] = lim f(x) ± lim g(x)

Sum/diff/product/quotient when each limit exists (denominator ≠ 0)

Constant Multiple

lim k·f(x) = k · lim f(x)

Pull out constants

Key Limits

Trig Basic Limit

lim_x→0 sin xx = 1

Essential — uses radians

Natural Exponential Limit

lim_x→0 e^x - 1x = 1

Derived from the definition of e

Natural Log Limit

lim_x→0 ln(1 + x)x = 1

From the previous limit by substitution

Wrap-up

Definition

lim_x→a f(x) = L ⟺ left limit = right limit = L

Both sides must agree on the same value

🎯 Exam Points

①If left ≠ right, the limit does not exist

②f(a) and the limit are independent — limit can exist even when f(a) is undefined

③Use factoring/rationalization/L'Hôpital for 0/0 forms

④Memorize sin x/x → 1 and (e^x − 1)/x → 1

⑤Limit arithmetic requires each limit to exist

Worked Examples & Past Exam

Example 1

Evaluate lim_x→2 (x² − 4)/(x − 2).

It is a 0/0 form, so factor the numerator and cancel.

x² - 4x - 2 = (x+2)(x-2)x-2 = x + 2

Substitute x → 2.

lim_x→2 (x + 2) = 4

▸ 4

Resolve a 0/0 form by factoring and cancelling, then substitute.

Example 2

Evaluate lim_x→0 (sin 3x)/x.

To use sin x / x → 1, match the denominator to 3x.

sin 3xx = 3 × sin 3x3x

As 3x → 0, sin(3x)/(3x) → 1.

3 × 1 = 3

▸ 3

To use lim_x→0 sin(x)/x = 1, make the denominator equal to the argument of sin.

2023 CSAT Math type, adapted

What is lim_x→1 (x² + 2x − 3)/(x − 1)?

①4

②0

③2

④3

⑤no limit

▸ ① 4

Factor the numerator: x² + 2x − 3 = (x + 3)(x − 1).

x² + 2x - 3x - 1 = (x+3)(x-1)x-1 = x + 3

Substitute x → 1.

lim_x→1 (x + 3) = 4

Continuity

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