Step by step
The question
Question: find the limit as x → 2 of this f(x): 6 − x if x ≤ 2, and x² if x > 2.
1 From the left
From the left means x is a bit less than 2.
There, f(x) is 6 − x.
So we use the first piece.
A polynomial piece has no gaps, so we just put 2 in.
We get 4.
2 From the right
From the right means x is a bit more than 2.
There, f(x) is x².
So we use the second piece.
A polynomial piece has no gaps, so we just put 2 in.
We get 4.
3 Compare the sides
From the left we get 4.
From the right we get 4.
Both sides give the same number.
So the limit is 4.
Careful!
One side is not enough.
A limit needs both sides to give the same number.
4 Check
Let's check on the graph.
The first piece comes in from the left.
It reaches height 4.
The second piece starts at height 4.
They meet: no jump.
At x = 2 both pieces arrive at 4.
Right!
Let's look back.
First: from the left.
Then: from the right.
Then: compare the sides.
Last: check.