Step by step
The question
Question: find the limit as x → 3 of 1/(x − 3).
1 Put the number in
Put 3 in place of x.
The top is 1, and the bottom is 0.
A number over 0 has no value.
And it is not 0/0, so there is nothing to cancel.
Careful!
1/0 is not 0.
We look at each side instead.
2 From the left
Numbers just below 3: 2.9, 2.99, 2.999.
The bottom is a tiny negative number.
So the values get bigger and bigger, below 0.
They go down without end.
We write −∞: minus infinity.
3 From the right
Numbers just above 3: 3.1, 3.01, 3.001.
The bottom is a tiny positive number.
So the values get bigger and bigger.
They go up without end.
We write ∞: infinity.
4 Compare the sides
From the left we get −∞.
From the right we get ∞.
The two sides do not agree.
So the limit does not exist.
5 Check
Let's check on the graph.
The dashed line is x = 3.
The graph never touches it.
Left of it, the graph drops down.
Right of it, the graph shoots up.
Right!
Let's look back.
First: put the number in.
Then: from the left.
Then: from the right.
Then: compare the sides.
Last: check.