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Wednesday, October 3, 2012

Indeterminate Form - Zero Over Zero

Category: Differential Calculus

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Evaluate 
 
 
Solution:

Examine first the given equation if there is a common factor. Since there's no common factor, then we can proceed to the limit. Substitute x = 2 to the given equation, we have
 

Any number, except zero, divided by zero is infinity (∞). Since 0/0 is not acceptable, then we have to apply the L'Hopital's Rule.Take the derivative of the numerator and denominator with respect to x,
 
 
 
Since their common factor is x at the numerator, then we can divide both sides by x, we have

 
Substitute x = 2 to the above equation, 

 
Therefore,
 
 
Note:

The L'Hopital's Rule is applicable for Indeterminate Forms like 0/0 and ∞/∞. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published the rule in his book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (literal translation: Analysis of the Infinitely Small for the Understanding of Curved Lines) (1696), the first textbook on differential calculus.