L'Hospital Rule

Guillaume de l'Hôpital, 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 ∞

L'hospital rule example 1

 

 

 

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.

L'hospital rule example 2

 

 

 

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.

L'hospital rule example 3

 

 

 

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.

L'hospital rule example 4

 

 

 

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.

L'hospital rule - sin(x) /x

 

 

 



 

 

 

 

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.
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