Step by step
The question
Question: find the limit as x → 4 of 1/(x − 4)².
1 Put the number in
Put 4 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 4: 3.9, 3.99, 3.999.
The bottom is a tiny positive number.
So the values get bigger and bigger.
They go up without end.
We write ∞: infinity.
3 From the right
Numbers just above 4: 4.1, 4.01, 4.001.
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 ∞.
Both sides go up without end, the same way.
So the limit is ∞: f(x) grows without end.
5 Check
Let's check on the graph.
The dashed line is x = 4.
The graph never touches it.
Left of it, the graph shoots up.
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.