Indeterminate Form - Combined, 2

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Evaluate

Solution:

To get the value of a given function, let's substitute the value of n to the above equation, we have  

Since the answer is (∞ - ∞)/∞ , then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We cannot use the L'Hopital's Rule because the Indeterminate form is (∞ - ∞)/∞. L'Hopital's Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. We have to do something first in the given equation so that the Indeterminate Form becomes 0/0 or ∞/∞.

Let's rewrite the above equation by expanding and simplifying the numerator as follows 

 

Substitute the value of n to the above equation, we have

Since the Indeterminate Form is  ∞/∞, then we can use the L'Hopital's Rule to the above equation as follows 

Since the variables at the right side of the equation are all canceled, then there's no other way to substitute the value of a variable to the right side of the equation. 

Therefore,

 

Indeterminate Form - Zero Times Infinity, 2

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate:

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have

Since the answer is 0•∞, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We cannot use the L'Hopital's Rule because the Indeterminate form is 0•∞. L'Hopital's Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. We have to do something first in the given equation so that the Indeterminate Form becomes 0/0 or ∞/∞.

Let's rewrite the above equation as follows

Substitute the value of x to the above equation, we have

Since the Indeterminate Form is  ∞/∞, then we can use the L'Hopital's Rule to the above equation as follows

 
Substitute the value of x to the above equation, we have

  
Therefore,

In this case, the above equation has no limit. 

Indeterminate Form - Zero Over Zero, 6

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have  

Since the answer is 0/0, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We have to do something first in the given equation so that the final answer will be a real number, rational, or irrational number.

Method 1:

Since the answer is Indeterminate Form, then we have to use the principles of trigonometric identities. Let's convert the double angle function into single angle function and then simplify as follows

Substitute the value of x to the above equation, we have 

Therefore,

Method 2:


Another method of solving Indeterminate Form is by using L'Hopital's Rule. This is the better method especially if the rational functions cannot be factored. L'Hopitals Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given function by taking the derivative of numerator and denominator with respect to x as follows

Substitute the value of x to the above equation, we have

Therefore,