Indeterminate Form - Zero Over Zero

Category: Differential Calculus

"Published in Newark, California, USA"
                     
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.

Squaring - Three Digit Number

Category: Arithmetic, Algebra

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Find the value ofwithout using a calculator.

Solution:

The given number is close to 100 which is a perfect square. We can write 103 as 100 + 3. By using the squaring of a binomial, we can square 103 without using a calculator as follows

Converting Triple Angle - Single Angle Formula

Category: Trigonometry

"Published in Suisun City, California, USA"
   
Convert into a single angle formula.

Solution:

Consider the following formulas:

Therefore,