Step by step
The question
Question: find the limit as x → 1 of this f(x): 3x if x < 1, and x + 5 if x ≥ 1.
1 From the left
From the left means x is a bit less than 1.
There, f(x) is 3x.
So we use the first piece.
A polynomial piece has no gaps, so we just put 1 in.
We get 3.
2 From the right
From the right means x is a bit more than 1.
There, f(x) is x + 5.
So we use the second piece.
A polynomial piece has no gaps, so we just put 1 in.
We get 6.
3 Compare the sides
From the left we get 3.
From the right we get 6.
The two sides do not agree.
So the limit does not exist.
Careful!
One side is not enough.
3 from the left and 6 from the right do not agree.
4 Check
Let's check on the graph.
The first piece comes in from the left.
It reaches height 3.
The second piece starts at height 6.
There is a jump.
At x = 1 the graph jumps from 3 to 6.
Right!
Let's look back.
First: from the left.
Then: from the right.
Then: compare the sides.
Last: check.