Limit — Precalculus, what it means and where it is taught
The value a function approaches as the input approaches some number.
Taught in Precalculus.
Where it is taught
What it rests on
Work through these first if this one does not settle.
- Function — A rule pairing each input with exactly one output.
What uses it later
This idea is needed for:
- Indeterminate form — 00, which means the limit needs more work — not that it fails to exist.
Practise this idea
8 questions in the course bank use Limit, in Geometric Construction, Series & Sigma Notation, Limits & Continuity.
Problem 1
If you are given only the three angle measures of a triangle, how many different triangles fit?
- Exactly one
- Exactly two
- None
- Infinitely many
Worked answer
Infinitely many
Angles fix the shape but not the size.
Every triangle with those angles is similar to the others, and there is no limit on how large you make it.
So infinitely many — this is why three angles do not determine a unique triangle, while three sides do.
Problem 2
Which infinite series has a finite sum?
- 1+12+14+⋯
- 1+2+4+⋯
- 1+1+1+⋯
- 2+4+8+⋯
Worked answer
1+12+14+⋯
Only the first has |r|<1
The others grow without limit
Problem 3
What does limx → a f(x) = L tell you about f(a)?
- Nothing
- That f(a)=L
- That f(a) is undefined
- That f(a) ≠ L
Worked answer
Nothing
The single point x=a is deliberately ignored
Problem 4
Substituting gives 00. What does that mean?
- More work is needed
- The limit is 0
- The limit is 1
- The limit does not exist
Worked answer
More work is needed
It is an indeterminate form
Top and bottom share a factor that has not been cancelled
Problem 5
For f(x)=2x when x<1 and f(x)=x+5 when x ≥ 1, find the limit as x approaches 1 from the left.
Worked answer
2
2(1) = 2
Problem 6
For the same function, find the limit as x approaches 1 from the right.
Worked answer
6
1+5 = 6
Problem 7
For that same function, does limx → 1 f(x) exist?
- No — the two sides disagree
- Yes, it is 2
- Yes, it is 6
- Yes, it is 4
Worked answer
No — the two sides disagree
2 from the left and 6 from the right
They must agree for the limit to exist
Problem 8
A function has a hole at x=2 but heads for 5 from both sides. Which is true?
- The limit is 5 but it is not continuous there
- It is continuous
- The limit does not exist
- The limit is 0
Worked answer
The limit is 5 but it is not continuous there
The limit exists; the value does not