lim · Limit
The limit asks where nearby values tend, independently of the value at the point.
0.9, 0.99, 0.999, … if you keep attaching 9s forever, is the end of that road 1 or not?
Distinguish the value at a point from the value approached nearby.
Begin by reading inputs and outputs in a table or graph.
The 2,000-year-old riddle of why Achilles can't catch the tortoise.
Open the immersive version →A reading path containing this lesson: Instantaneous change and accumulation · 1/3
We can read nearby values even when a point is missing
Consider f(x)=(x²−4)/(x−2) for x≠2. Factoring the numerator gives (x−2)(x+2). When x is not 2, cancellation gives x+2.
Input 1.9 gives 3.9, and 1.99 gives 3.99. From the other side, 2.01 gives 4.01. As the input approaches 2, the output approaches 4.
But the original expression is undefined at 2 because the denominator is zero. The value at 2 and the value approached near 2 are different questions.
Saying the limit is 4 is stronger than checking a few numbers. For nearby x≠2, |f(x)−4|=|x−2|. Making the input error small makes the output error just as small.
Must a function never reach its limit? No. The constant function f(x)=1 already equals 1 everywhere and has limit 1 at every point. Nearby behaviour, not failure to reach a value, is the key.
We could set the value at 2 separately to 1 while keeping x+2 nearby. The limit at 2 would still be 4. Read the point and its neighbourhood separately in the picture and experiment.
Read these symbols together
- limₓ→ₐ f(x)=L
- As x approaches a, f(x) can be made arbitrarily close to L. Here we consider finite real limits.
- 0 < |x−a|
- We look at inputs near a but different from a itself.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — is 0.999… (9s repeating forever) a bit less than 1, or exactly 1?
Attach them yourself — one 9 at a time
Current value
0.9
Gap to 1
0.1
A value at a point and a value approached
A limit is a destination, not an arrival.
The metaphor says reaching the limit is not required, not that it is forbidden. The constant function f(x)=1 takes its limit value at every point. The constant sequence aₙ=1 also equals its limit at every term.
Undefined division 0/0 and a 0/0-type limit are different questions.
Division 0/0 is undefined over the reals. A limit whose numerator and denominator both tend to zero is a separate question. As x→0, x/x→1 and x²/x→0, while |x|/x has no two-sided limit. The same indeterminate form can lead to different outcomes.
Express closeness precisely
Written in the language of mathematics, the approach experiment says:
Limit notation
As x gets endlessly close to a, f(x) gets endlessly close to L. You're not substituting x=a — you're reading the surroundings' destination.
The famous limit from the experiment
Here x is measured in radians. The quotient is undefined at 0, while its limit as x→0 is 1. With this convention, sin x ≈ x for small x.
What 0.999… = 1 really is
The notation 0.999… is itself a number defined by a limit. Since that limit is 1, 0.999… simply is 1.
The sin x/x example uses radians. Finite numerical trials do not prove a general limit.
References
- OpenStax · Calculus Volume 1 · 2.3 Limit Laws
Make one more distinction
0.999… is a tiny bit less than 1.
No number can fit between 0.999… and 1 — the difference is 0, and two numbers whose difference is 0 are the same number. 0.999… = 1 is not an approximation; it's an exact equality.
A limit only exists if the function has a value at that point.
A limit looks only at the surroundings. Even if the point has a hole (undefined), or a completely wrong value, the limit exists as long as the surroundings point to one place. This separation is what made derivatives possible.
Where else can this idea help?
The foundation of calculus
Instantaneous speed (derivative) and area under a curve (integral) are both defined by limits. Without limits, the entire building of calculus cannot stand.
Curves on your screen
Computers draw curves as many very short straight lines. Shrink the pieces endlessly and it becomes indistinguishable from a curve — rendering is limits as engineering.
The limit of interest: e
Compound interest in infinitely small slices (continuous compounding) produces e = 2.718…, the limit of (1+1/n)ⁿ. The natural constant is a number born from a limit.
Convergence and numerical error
Numerical methods examine how approximations behave as a step size shrinks. An error guarantee needs model assumptions and separate error analysis. Structural limit-state design is not the same concept merely because it has a similar name.
Explain it in your own words
Compare with one possible explanation
A table checks only those inputs. A proof must control all sufficiently close inputs for any prescribed positive tolerance.
Try another question
Q1lim(x→3) (x²−9)/(x−3) = ? (Substituting x=3 gives 0/0. Look at the 'surroundings' like in the experiment.)Show answer ▾
6. Factor the numerator: (x−3)(x+3)/(x−3) = x+3, and as x approaches 3, x+3 approaches 6. Exactly the same structure as the function-with-a-hole experiment.
Q2Is 0.333… (3s repeating forever) exactly equal to 1/3, or a tiny bit smaller?Show answer ▾
Exactly equal. Same logic as 0.999…=1 — the notation 0.333… is itself a number defined as a limit, and that limit is 1/3.
Q3What is the limit of the sequence 1/2, 1/4, 1/8, 1/16…? Which term of the sequence actually reaches that limit?Show answer ▾
The limit is 0, but no term ever becomes 0. The destination is fixed even though it's never reached — that's the core intuition of limits.
Try answering yourself before revealing it — getting it wrong is where learning starts.
The idea to take with you
Separating the point value from nearby behaviour reveals the distinction between limits and continuity.
What question comes next?
A reading path containing this lesson: Instantaneous change and accumulation · 1/3
Through a university lens
The ε-δ Definition
You've felt 'arbitrarily close' with your hands — now turn that phrase into the rules of a game. University analysis starts here.
Go deeper →
Previous concept
Sequences
Where does a sequence go if it continues forever — the question of limits was first born from sequences.
← Sequences labLeads to next
Derivatives
Now that the tool of limits is in hand, let's do the real work of 'sending the interval to 0'. The instantaneous rate of change — the derivative.
Go to the Derivatives lab →Related lab
Integrals
The definition of the integral — 'rectangles infinitely thin' — is a limit. Watch a limit build an area.
Go to the Integrals lab →Related lab
Compound Interest
Where do you arrive when interest is compounded in infinitely small slices? Meet the limit that gave birth to e, in finance.
Go to the Compound Interest lab →