Step by step
The question
Question: decide if f(x) = 3/(x + 1) is continuous at x = −1.
f(x) = 3x + 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. f(−1) has a value
2. the limit at −1 exists
3. limit = f(−1)
1 Does f have a value?
Put −1 in place of x.
f(−1) = 3(−1) + 1 = 30
The bottom is 0.
We cannot divide by 0.
So f(−1) has no value.
a number over 0: no value
Check 1 is false.
We can stop here: one false check is enough.
1. f(−1): no value ✗
So f is not continuous at x = −1.
= f is not continuous at x = −1
2 Check
Let's check on the graph.
At x = −1 the graph shoots off: it never reaches x = −1.
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 = −1