Limits (Evaluating)
You should read Limits (An Introduction) first
Quick Summary of Limits
Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer!For example: | (x^{2}-1)/(x-1) |
At x=1: | (1^{2}-1)/(1-1) = (1-1)/(1-1) = 0/0 |
But 0/0 is "indeterminate", meaning we can't determine its value. But instead of trying to work it out for x=1 let's try approaching it closer and closer:
x | (x^{2}-1)/(x-1) |
0.5 | 1.50000 |
0.9 | 1.90000 |
0.99 | 1.99000 |
0.999 | 1.99900 |
0.9999 | 1.99990 |
0.99999 | 1.99999 |
... | ... |
Now we can see that as x gets close to 1, then (x^{2}-1)/(x-1) gets close to 2
We are now faced with an interesting situation:
- When x=1 we don't know the answer (it is indeterminate)
- But we can see that it is going to be 2
We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"
The limit of (x^{2}-1)/(x-1) as x approaches 1 is 2
And it is written in symbols as:
So it is a special way of saying, "ignoring what happens when you get there, but as you get closer and closer the answer gets closer and closer to 2"
As a graph it looks like this: So, in truth, you cannot say what the value at x=1 is. But you can say that as you approach 1, the limit is 2. |
Evaluating Limits
"Evaluating" means to find the value of (think e-"value"-ating)
In the example above we said the limit was 2 because it looked like it was going to be. But that is not really good enough!
In fact there are many ways to get an accurate answer. Let's look at some:
1. Just Put The Value In
The first thing to try is just putting the value of the limit in, and see if it works (in other words substitution).
Let's try some examples:
Example | Substitute Value | Works? | |
---|---|---|---|
(1-1)/(1-1) = 0/0 | |||
10/2 = 5 |
It didn't work with the first one (we knew that!), but the second example gave us a quick and easy answer.
2. Factors
You can try factoring.
Example: | |
By factoring (x^{2}-1) into (x-1)(x+1) we get: | |
Now we can just substitiute x=1 to get the limit: | |
3. Conjugate
If it's a fraction, then multiplying top and bottom by a conjugate might help.
The conjugate is where you change the sign in the middle of 2 terms like this: |
Here is an example where it will help you to find a limit:
Evaluating this at x=4 gives 0/0, which is not a good answer! |
So, let's try some rearranging:
Multiply top and bottom by the conjugate of the top: | ||
Simplify top using : | ||
Simplify top further: | ||
Eliminate (4-x) from top and bottom: |
So, now we have:
Done!
4. Infinite Limits and Rational Functions
A Rational Function is one that is the ratio of two polynomials: | ||
For example, here P(x)=x^{3}+2x-1, and Q(x)=6x^{2}: |
By finding the overall Degree of the Function we can find out whether the function's limit is 0, Infinity, -Infinity, or easily calculated from the coefficients.
Read more at Limits To Infinity.
5. Formal Method
The formal method sets about proving that you can get as close as you want to the answer by making "x" close to "a".
Read more at Limits (Formal Definition)