Step by step
The question
Question: decide if f(x) = 1/(x − 2) is continuous at x = 2.
f(x) = 1x − 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. f(2) has a value
2. the limit at 2 exists
3. limit = f(2)
1 Does f have a value?
Put 2 in place of x.
f(2) = 12 − 2 = 10
The bottom is 0.
We cannot divide by 0.
So f(2) has no value.
a number over 0: no value
Check 1 is false.
We can stop here: one false check is enough.
1. f(2): no value ✗
So f is not continuous at x = 2.
= f is not continuous at x = 2
2 Check
Let's check on the graph.
At x = 2 the graph shoots off: it never reaches x = 2.
We must lift the pen to draw it.
Right!
Let's look back.
First: does f have a value?.
Last: check.
Answer
= f is not continuous at x = 2