There are many ways to get an accurate answer. Let's look at some:
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).
We can try factoring.
If we simply subsitute x=1 into this, we will get 0/0 which is undefined.
By factoring (x2−1) into (x−1)(x+1) we get:
Now we can just substitiute x=1 to get the limit:
1+1 = 2
For some fractions multiplying top and bottom by a conjugate can help.
The conjugate is where we change the sign in the middle of 2 terms like this:
Here is an example where it will help us 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:
Cancel (4−x) from top and bottom:
So, now we have:
Infinite Limits and Rational Functions
A Rational Function is one that is the ratio of two polynomials
For example, here P(x) = x3 + 2x − 1, and Q(x) = 6x2:
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.
Learn about the limits to infinity here.
Drafted by Eunice (Maths)