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) = 0.
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 0.
2 Does the limit exist?
Near −3, but not at −3, f(x) is (x² − 9)/(x + 3).
Factor the top and the bottom.
x is never −3 here, so x + 3 is not 0.
Cross it out on the top and the bottom.
Put −3 in what is left.
The limit is −6.
Check 2 is true: the limit exists.
3 Are they equal?
The limit is −6, but f(−3) is 0.
They are not equal.
Check 3 is false.
So f is not continuous at x = −3.
4 Check
Let's check on the graph.
The graph comes to height −6 from both sides, but there is a hole at x = −3.
f(−3) = 0 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.