Step by step
The question
Question: find the limit as x → 9 of (√x − 3)/(x − 9).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Put the number in
Put 9 in place of x.
The top is 0, and the bottom is 0.
The square root of 9 is 3.
3 minus 3 is 0.
0/0 has no value.
It tells us to work more.
2 Multiply by the conjugate
The conjugate of √x − 3 is √x + 3: the same two terms, with the sign swapped.
Multiply the top and the bottom by the conjugate.
Why?
A minus times a plus squares the root, and a squared root has no root left.
Something over itself is 1.
So the value does not change.
Careful!
Both the top and the bottom!
One alone would change the value.
3 Multiply out
Remember: a minus b, times a plus b, is a squared minus b squared.
The middle terms cancel.
On top: (√x − 3) times (√x + 3).
That is the root squared, minus 3 squared.
The bottom we leave as it is.
A square root squared is just what is inside.
So it is x − 9.
Now (x − 9) is on the top and on the bottom.
4 Cancel
x gets close to 9, but it is never 9.
So x − 9 is not 0.
Something over itself is 1.
So we cross out (x − 9) on the top and the bottom.
Write what is left: 1 over (√x + 3).
5 Put the number in again
Now the 0 is gone.
Put 9 in again.
The square root of 9 is 3.
3 + 3 is 6.
So the limit is 1/6.
6 Check
Let's check with numbers close to 9.
A calculator gives 0.1667 at x = 8.99.
And 0.1666 at x = 9.01.
Both are close to 1/6, which is 0.1667.
Right!
7 Pick the choice
Our answer is limit = 1/6.
That is choice A.
8 Why not the others
B is 0.
Wrong: 0/0 means simplify first.
C is 6.
Wrong: turned the fraction upside down.
D is 1/3.
Wrong: 1 over √9: lost the + 3.
Let's look back.
First: put the number in.
Then: multiply by the conjugate.
Then: multiply out.
Then: cancel.
Then: put the number in again.
Then: check.
Then: pick the choice.
Last: why not the others.