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 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?
Even when you cannot plug a number into a point, you can still ask where nearby outputs settle. That question is the limit of a function, and it is why a 0/0 form is still worth organizing instead of throwing away. Left-hand and right-hand limits have to match before you name a single limit, because one side alone can sit at another height. Continuity is a later comparison between that limit and f(a), so the approach comes first. Sums, differences, products, and quotients of limits are computed from the separate limits, and a quotient needs a nonzero limit in the denominator.
🎯 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
④A limit can exist even if f(a) does not. L'Hôpital is in Calculus
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Undefined at x = 1 yet converges to 2
The approach sketch draws f(x) = (x² − 1)/(x − 1) in green, with a red hollow circle at height 2 on x = 1. A blue point sits on the left of the curve and an orange point on the right, and dashed lines mark the hole at x = 1 and y = 2. Closeness t runs from 0 % to 95 %. A larger t places both points nearer the hole. The middle sketch is f(x) = 2x with limit 4 as x → 2. The orange band is ε around L = 4 and the blue band is δ around a = 2; on this line they are paired as δ = ε/2. ε runs from 0.3 to 3.0. The lower sketch overlays green f(x) = x²/3 and orange k · f(x) to show the constant-multiple rule, with k from 0.5 to 3.0.
💡 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 (advanced)
The red hollow circle is easy to read as “no limit.” That circle only marks a missing output at x = 1; it is not a verdict on the limit. On this figure the blue and orange points take new seats toward the same height 2, so a hole can still leave one limit candidate. Having a number for f(a) also does not make that number the limit by itself. If the two sides sit at different heights, there is no limit even when f(a) exists, and continuity fails then as well. Plugging the target into a 0/0 form collapses the expression, so you first look for a factor you can cancel.
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Smaller ε automatically shrinks δ — the rigorous definition
Definition (ε-δ)
limx→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
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Constant multiple — foundation of limit arithmetic
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
limx→0sin xx = 1
Calculus scope. Uses radians. Not required for Math II exams
Natural Exponential Limit
limx→0ex - 1x = 1
Calculus scope. Derived from the definition of e
Natural Log Limit
limx→0ln(1 + x)x = 1
Calculus scope. From the previous limit by substitution
Wrap-up
Definition
limx→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 limx→2 (x² − 4)/(x − 2).
1
It is a 0/0 form, so factor the numerator and cancel.
x2 - 4x - 2 = (x+2)(x-2)x-2 = x + 2
2
Substitute x → 2.
limx→2 (x + 2) = 4
▸ 4
Resolve a 0/0 form by factoring and cancelling, then substitute.
Example 2
Evaluate limx→0 (sin 3x)/x.
1
To use sin x / x → 1, match the denominator to 3x.
sin 3xx = 3 × sin 3x3x
2
As 3x → 0, sin(3x)/(3x) → 1.
3 × 1 = 3
▸ 3
To use limx→0 sin(x)/x = 1, make the denominator equal to the argument of sin.