The Limit of a Sequence
โA limit is not 'arrival' but the value you draw endlessly close to.โ
The formula
lim(nโโ) aโ = LHow to read it: as n grows without bound, if the sequence value aโ gets arbitrarily close to a single value L, then L is the limit of the sequence
- aโ
- โ the n-th value of the sequence
- n โ โ
- โ the term number n growing without bound
- L
- โ the limit the sequence approaches
The hook
1, 1/2, 1/3, 1/4, โฆ never becomes 0, yet it draws 'endlessly' close to 0 โ that destination is the limit.
In plain words
As you push the term number n to infinity, if the terms keep getting closer to some single number L, then L is the limit of the sequence.
The intuition
Picture steps that forever approach a destination but never step on it. What matters isn't 'do you arrive' but 'can you get as close as you like'. If no matter how tiny an error you demand, going far enough gets you within it, then that approached value is the limit.
How it's built
lim(nโโ) means 'as n grows without bound', aโ is the sequence, L is the approached value. If the limit settles on one finite value it 'converges'; if it grows without bound or never settles, it 'diverges'.
Example
aโ = 1/n runs 1, 1/2, 1/3, โฆ , getting ever smaller. As n grows it draws endlessly close to 0, so lim(nโโ) 1/n = 0.
Common mistake
A limit does NOT mean you 'actually reach' the value. 1/n is never 0 for any n โ it only gets as close to 0 as you like. 'Approaching' and 'equaling' are different.
Where it's used
Approximating a circle's area with polygons, summing infinite series, the foundation of derivatives and integrals โ the starting point of all of calculus, 'slicing infinitely fine and approaching'.
Where it came from
Pinning down 'endless approach' in the rigorous ฮต-N language was the work of Cauchy and Weierstrass in the 1800s, which set calculus on solid logical ground.
Prerequisites
Quick check
What is lim(nโโ) 1/n?
- 0โ
- 1
- โ
- 1/2
Practice
Find lim(nโโ) 1/nยฒ.
Answer: 0
- As n grows, nยฒ grows even faster
- 1/nยฒ draws endlessly close to 0 โ 0
Takeaway: When the denominator blows up, the fraction goes to 0.
Find lim(nโโ) (2n+1)/n.
Answer: 2
- (2n+1)/n = 2 + 1/n
- As nโโ, 1/n โ 0, so the limit is 2
Takeaway: Splitting the expression reveals the vanishing term.
For the constant sequence aโ=3, find lim(nโโ) 3.
Answer: 3
- Every term is always 3
- The value never changes, so the limit is 3
Takeaway: A constant sequence's limit is that constant.
Explain what it means that 1/n approaches 0 but never becomes 0.
Answer: undefined
- For any n, 1/n > 0
- But by growing n you get as close to 0 as you like
- The limit 0 is not a 'reached point' but the 'approached value'
Takeaway: A limit is infinite approach, not arrival.