Let's learn how to find the value with limits in A-Level Maths!
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).
Example:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/989842_498519.jpeg)
10/2 =5
Factors
We can try factoring.
Example:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/969091_638616.jpeg)
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:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/712353_339997.png)
Now we can just substitiute x=1 to get the limit:
1+1 = 2
Conjugate
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:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/225274_690703.png)
Here is an example where it will help us find a limit:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/772031_212698.jpeg)
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:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/811582_378154.png)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/270604_526407.png)
Cancel (4−x) from top and bottom:
So, now we have:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/177938_347937.png)
Infinite Limits and Rational Functions
A Rational Function is one that is the ratio of two polynomials
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/258180_750787.jpeg)
For example, here P(x) = x3 + 2x − 1, and Q(x) = 6x2:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/207800_736756.jpeg)
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)
Reference
https://www.mathsisfun.com/calculus/limits-evaluating.html