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,

 

Indeterminate Form - Infinity Over Infinity, 3

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 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 divide both sides of the fraction by a term with the highest degree which is n³ and simplify the given equation as follows

Substitute the value of n 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 n as follows  

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

Therefore,

 

Indeterminate Form, Infinity Over Infinity, 2

Category: Differential Calculus, Algebra

"Published in Vacaville, 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 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 divide both sides of the fraction by 2ⁿ and simplify the given equation as follows

Substitute the value of n 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 n as follows 

Again, apply the L'Hopital's Rule to the above equation, we have

Since the resulting equation is the same as the given equation, then we cannot use the L'Hopital's Rule because of the repetitive solutions and results. There's no end in this process. In this case, we have to consider the Method 1 in solving the given limits.  

Indeterminate Form, Zero Over Zero, 5

Category: Differential Calculus, Algebra

"Published in Vacaville, 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 inspect the given function, factor, and then simplify if possible 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,