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