Step by step
The question
Question: decide if this f(x) is continuous at x = 3.
Continuous at x = 3 means three things are true.
One: f(3) has a value.
Two: the limit there exists.
Three: they are equal.
1 Does f have a value?
Which piece holds x = 3?
The one with x ≥ 3.
So f(3) = 5.
Careful!
Having a value is not enough.
The graph must also arrive at that height from both sides.
Check 1 is true: f(3) is 5.
2 Does the limit exist?
From the left, we use 2x − 1.
We get 5.
From the right, we use x + 2.
We get 5.
Both sides give 5.
Check 2 is true: the limit exists.
3 Are they equal?
The limit is 5, and f(3) is 5.
They are equal.
Check 3 is true.
All three checks are true.
So f is continuous at x = 3.
4 Check
Let's check on the graph.
The first piece comes in from the left.
The second piece goes on from the same point.
No lifting the pen.
Right!
Let's look back.
First: does f have a value?.
Then: does the limit exist?.
Then: are they equal?.
Last: check.