The idea of a limit
Getting close
Can a function have a limit at a point where it is not even defined? Can it be defined at a point and still have no limit there?
A limit is a prediction. It asks which value f(x) is heading toward as x gets close to a, without ever using x=a. Move the ochre point toward a, read the table, and make your own estimate before checking it.
Drag the ochre point along the curve, or use the controls. The point never lands on a itself.
| Distance from a | f(a − distance) | f(a + distance) |
|---|
What the notation says
We write
when the values f(x) can be made as close to L as we like by taking x close enough to a, on both sides, with x\ne a. Three things follow from that sentence, and each one matches a function in the graph above.
- The value f(a) does not matter
The limit uses only x\ne a. Compare the hole with the stray point: what is the limit, and what is f(1)?
- Both sides must agree
If f(x) heads to one value from the left and another from the right, there is no single prediction, so the limit does not exist. Try the jump.
- The values must settle
If f(x) grows without bound, or keeps swinging between values, it never settles on a number. Try the spike and the oscillation.
Tables and graphs give strong evidence, but not proof. The oscillating function fools any table that samples only certain values of x. The lessons that follow show how to find limits exactly.
Solve it step by step
Read each limit from the graph. Type a number, inf or -inf for values that grow without bound, or dne when something does not exist. The score counts steps you answer correctly on the first try.