Step by step
The question
Question: decide if f(x) = (x² − 9)/(x − 3) is continuous at x = 3.
f(x) = x2 − 9x − 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. f(3) has a value
2. the limit at 3 exists
3. limit = f(3)
1 Does f have a value?
Put 3 in place of x.
f(3) = 32 − 93 − 3 = 00
The bottom is 0.
We cannot divide by 0.
So f(3) has no value.
0 over 0: no value
Careful!
The limit does exist here: it is 6.
But f(3) has no value.
The graph has a hole.
Careful: the limit 6 is not enough
Check 1 is false.
We can stop here: one false check is enough.
1. f(3): no value ✗
So f is not continuous at x = 3.
= f is not continuous at x = 3
2 Check
Let's check on the graph.
The graph comes to height 6 from both sides, but there is a hole at x = 3.
We must lift the pen.
Right!
Let's look back.
First: does f have a value?.
Last: check.
Answer
= f is not continuous at x = 3