Step by step
The question
Question: decide if this f(x) is continuous at x = 2.
Continuous at x = 2 means three things are true.
One: f(2) has a value.
Two: the limit there exists.
Three: they are equal.
1 Does f have a value?
Which piece holds x = 2?
The one with x = 2.
So f(2) = 5.
Careful!
Having a value is not enough.
The graph must also arrive at that height from both sides.
Check 1 is true: f(2) is 5.
2 Does the limit exist?
Near 2, but not at 2, f(x) is (x² − 4)/(x − 2).
Factor the top and the bottom.
x is never 2 here, so x − 2 is not 0.
Cross it out on the top and the bottom.
Put 2 in what is left.
The limit is 4.
Check 2 is true: the limit exists.
3 Are they equal?
The limit is 4, but f(2) is 5.
They are not equal.
Check 3 is false.
So f is not continuous at x = 2.
4 Check
Let's check on the graph.
The graph comes to height 4 from both sides, but there is a hole at x = 2.
f(2) = 5 is a dot somewhere else.
We must lift the pen.
Right!
Let's look back.
First: does f have a value?.
Then: does the limit exist?.
Then: are they equal?.
Last: check.