Limits Near a Point → Explorer
Approach x = a from both sides and watch where the outputs are heading — whether or not the function has a value there.
Point L — approaching from the leftPoint R — approaching from the rightOpen circle — value not takenFilled dot — the value f(a) takesx = a
Live values
LFrom the
left
left
xL1.000
f(xL)1.500
RFrom the
right
right
xR3.000
f(xR)1.500
f(a) — the value at a
Defined at x = 2
1
Left-hand limit?
Right-hand limit?
Same destination from both sides??
Two-sided limit?
f(a)?
Move the points, predict the destination, then pressReveal conclusion.
Drag toward a
Jump to δ:
Numerical evidence
| δ | xL = a − δ | f(xL) | xR = a + δ | f(xR) |
|---|
A table gives evidence for a limit; it does notprove the limit.
- Left-hand limit
- Where the outputs head as x approaches a from values below a.
- Right-hand limit
- Where the outputs head as x approaches a from values above a.
- f(a)
- The output the function actually takes at a — if it takes one at all.
Key idea
- The points never reach a — a limit is about the approach, not the arrival
- A finite two-sided limit exists only when the left-hand and right-hand limits are equal
- The limit can exist where f(a) does not, andf(a) can exist without matching the limit
A limit asks where the function is going, not where it is.
A finite two-sided limit exists only when the left-hand and right-hand limits are equal.