### L'Hospital Rule

Guillaume de l'HÃ´pital, famously known as L'Hospital, came up with a method to deal with fractions, when they take the indeterminate forms, approaching certain limits.

The indeterminate form can be as follows:

•   0/0 - as in the case of sin(x)/x, when x approaches 0
•   ∞ / ∞ - as in the case of ex2 / x2, when x approaches ∞

L'Hospital Rule help us deal with situations of this kind. It is as follows:

#### L'Hospital Rule

If Limit[f(x)/g(x)] as x approaches a is 0/0 or ∞ / ∞, then Limit[f(x)/g(x)] as x approaches a is [f'(x)/g'(x)].

In all the following animations, [f(x)/g(x)] is drawn in red and [f'(x)/g'(x)] in purple.

Please note the convergence pf the two curves / lines to the same point, as the limit approaches.

E.g.1

Find Limit [(4x-3)/(5x-6)] as x approaches ∞.

If x = ∞, then [(4x-3)/(5x-6)] = ∞/∞ - indeterminate
Let's use L'Hospital rule for this:
f'(x)/g'(x) = 4/5 = 0.8, as x approaches ∞

E.g.2

Find Limit [(x-4)/ln(x-3)] as x approaches 4.

If x = 4;, then [(x-4)/ln(x-3)] = 0/0 - indeterminate
Let's use L'Hospital rule for this:
f'(x)/g'(x) = 1/(1/x-3) = 1, as x approaches 4.

E.g.3

Find Limit [ln(x)/√x] as x approaches ∞.

If x = ∞, then [ln(x)/√x] = ∞/∞- indeterminate
Let's use L'Hospital rule for this:
f'(x)/g'(x) = (1/x)/(1/2)x-1/2 = 2/√x = 0, as x approaches ∞.
The behaviour of the curve will be clearer when x is really large.

E.g.4

Find Limit [(x2 -x - 6)/(x2 -3x)] as x approaches 3;.

If x = 3, then [(x2 -x - 6)/(x2 -3x)] = 0/0 - indeterminate
Let's use L'Hospital rule for this:
f'(x)/g'(x) = (2x-1)/(2x-3) = 5/3, as x approaches 3.

E.g.5

Find Limit sin(x) /x as x approaches 0 and hence sketch y = sin(x)/x.

sin(x) /x when x approaches 0 = sin(0)/0 = 0/0 - indeterminate
Let's use L'Hospital rule for this:
f'(x) = cos(x); g'(x) = 1
So, f'(x)/g'(x) = cos(x)/1
When x approaches 0, f'(x)/g'(x) = 1/1 = 1
Therefore, sin(x)/x, when x approaches 0 = 1.

### Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

Email:

### Stand Out - from the crowd

"There's no such thing as a free lunch."

The best things in nature are free with no strings attached - fresh air, breathtakingly warm sunshine, scene of meadow on the horizon...

Vivax Solutions, while mimicking nature, offers a huge set of tutorials along with interactive tools for free.

Please use them and excel in the sphere of science education.

Everything is free; not even registration is required.

"