L'Hôpital's Rule can help us calculate a limit that may otherwise be hard or impossible. Let's learn about this in A-Level Maths.
It says that the limit when we divide one function by another is the same after we take the derivative of each function (with some special conditions shown later).
In symbols we can write:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/215087_312649.jpeg)
The limit as x approaches c of "f-of−x over g-of−x" equals the the limit as x approaches c of "f-dash-of−x over g-dash-of−x"
the ’ on each function means to take the derivative.
Example:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/975919_404931.jpeg)
if x=2, it will become 0/0
Now try L'Hôpital!
Differentiate both top and bottom:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/232963_293415.jpeg)
Now we just substitute x=2 to get our answer:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/75625_962890.jpeg)
Here is the graph, notice the "hole" at x=2:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/471813_322730.png)
Note: we can also get this answer by factoring, see here.
Conditions
Differentiable
For a limit approaching c, the original functions must be differentiable either side of c, but not necessarily at c.
Likewise g’(x) is not equal to zero either side of c.
The Limit Must Exist
This limit must exist as:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/114271_527446.jpeg)
example:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/47174_197151.jpeg)
differentiating top and bottom, 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/4979856/680493_360832.jpeg)
And because it just wiggles up and down it never approaches any value.
So that new limit does not exist!
And so L'Hôpital's Rule is not usable in this case.
BUT we can do this:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/502856_419841.jpeg)
As x goes to infinity then cos(x)/x tends to between −1/∞ and +1/∞, and both tend to zero.
And we are left with just the "1", so:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/358012_466688.jpeg)
Drafted by Eunice (Maths)
Reference
https://www.mathsisfun.com/calculus/l-hopitals-rule.html